What strategic value did Christianity hold for Constantine that made him convert to Christianity?

One theory advanced by the Emperor Julian (Constantine's nephew) and sole survivor (with his brother Constantius Gallus) of the wholesale massacre of the family initiated by the three surviving sons of Constantine, before they began murdering each other, was that Constantine was so haunted by his crimes that he had sought absolution through every available spiritual option of the day - after all he had murdered his eldest son Crispus and his wife Fausta in the most psychopathically cruel way imaginable, yet when he tried to apply for the Mysteries at Eleusis he was turned away because of his cruelty, the number of murders he had committed and the fact that he was irredeemably evil in the eyes of the gods.

But then along came Eusebius of Caesarea who whispered in the usurper's ear that he need only go through the motions of asking forgiveness of the God of the Jews and the spiritual entity that was known as Jesus Christ (this was before the church tried to create a historical Jesus) he would be cleansed of all his crimes and sins and be born anew!

Ammianus also repeats this rumour so it must've been fairly widely known - Constantine's reputation for extreme cruelty and as a tyrant and a serial killer certainly was - so presumably he turned to Christianity as means of dealing with his own conscience and the guilt he must have borne having so much blood on his hands, including members of his immediate family.

Constantine did not officially convert to Christianity until the very end of his life-(possibly on his deathbed). By all historical accounts, Constantine, was very much, (despite his alleged vision of The Cross), a Roman pagan, but, a Roman pagan who legalized Christianity in the city of Milan around 311-312 AD/ CE-(known as The Edict of Milan). Remember, Constantine was NOT responsible for nationalizing Christianity within the Roman Empire-(that would come decades later with the Byzantine Emperor, Theodosius and his rabidly anti-pagan based, Edict of Thessaloniki).

Although Constantine converted to Christianity at the very end of his life, he did spend much of his political tenure working with Christian communities throughout the Empire. Constantine, oversaw The Nicene Council, which officially discarded and nullified the centuries old diverse interpretations of the Jesus story and subsequently, united The New Testament into the "Synoptic Gospels"-(which exist to this day). Yet, despite Constantine's presence at this Landmark Council, he was still not officially, a Christian.

I do not have the exact statistical account as to how many Christians and pagans existed within the Italian peninsula and much of the Roman Empire during Constantine's time. However, the number of Christian communities were steadily multiplying since the 1st AD/CE and the demographic reality and presence of Christian communities living in Italy, as well as other parts of the Roman Empire, could no longer be ignored or dismissed as a fringe or peripheral minority. The arrest, torture and brutal execution of Christians was no longer an option for the Roman imperial government, due to the rising presence of Christian communities in Rome and elsewhere within the Empire.

So in terms of its possible "strategic value", Constantine's welcoming relations with the Christians would have, in the long run, helped to secure, insure and protect his political tenure. Had Constantine continued to persecute the Christians, the way his imperial predecessors had done, it is questionable as to whether or not he would have remained in power for 30 plus years. The persecution and literal "feeding the Christians to the lions" in the Coliseum, did not work for any of his predecessors and History shows, that such brutal actions had a counterproductive effect.

In other words, "at the end of the day", Constantine essentially "saw the Christian writing on the wall".......both symbolically and literally.

The Question as written, is a little bit leading. Constantine the Great converted to christianity on his deathbed and thus his motivation was not earthly and personal.

He was baptized, and received his first Communion on his death-bed, when sixty-four years old, and is remembered as the first believing monarch.

Constantine likewise as I recall converted to several religions at the same time on his deathbed although I can't source that, so his motivation likely wasn't exactly in line with Christian values either. It was more covering his bases in case their was a Christian afterlife, a jewish afterlife, etc..

If the question is why did Constantine in 313 AD, issue the Edict of Milan decriminalizing Christian worship.. It was certainly not to loot the christian temples, which is absurd since christianity as an outlawed underground movement had no such public temples or riches. As an outlawed persecuted religion such posessions would have long ago been confiscated if they had existed. Only, those possessions had not yet occurred, they would come as a result of being legitimized over centuries.

Constantine's decriminalization was more about forming an alliance.

Although the Edict of Milan is commonly presented as Constantine’s first great act as a Christian emperor, it is unlikely that the Edict of Milan was an act of genuine faith. The document should instead be seen as Constantine's first step in creating an alliance with the Christian God, who he considered the strongest deity.[10] At that time, he was more concerned about social stability and the protection of the empire from the wrath of the Christian God than he was for justice or care for the Christians. The edict is more indicative of Roman culture’s obsession with seeking the gods' intervention than of Constantine's or Licinius's religious beliefs.

313 AD was a data point on a long line of fascination with Christianity for Constantine. Constantine's interest in Christianity wasn't wealth, it was marshal. Constantine came to power by defeating rival roman emperors by rallying Christians to his cause. He as many Romans had witnessed Christians going to their deaths across the empire singing and rejoicing; and Constantine had thought to himself.. If I had a few legions of those guys I could deal with the fractured empires troubles. So Constantine began to recruit Christians for his army. The battle of Milvian Bridge also 312 AD, is an example of Constantine use of Christians as his warriors. Fighting under a Christian cross, and giving all credit for the victory to the Christian God, and representing himself as the chosen one of God to lead the empire. Only Constantine was still a Pagan when he did all these things.

The flaw in Constantine's plan was that before he could use christianity to unity his empire; first Constantine had to unify the Christian churches. He had to create a unified Church a catholic Church. The problem was in 325 AD christianity was a series of loosely affiliated underground religions which had survived for centuries of Roman persecution. Christians were put to death across the empire as they were discovered, so christians had become skilled at hiding. This hiding didn't allow the free flow of ideas. Christianity was thus a fractured religion made up of many different beliefs, each city lead by a bishop with his own doctrine which may or may not agree with the bishop who had set up shop just over the horizon in the next city. These bishops traditions and doctrines had evolved independently as their was no authority and limited contact with other branches of christianity. So Constantine couldn't use christianity to unit the empire until he united christianity. On this note, the pagan Constantine the Great called the first christian church council, the council of Nicea. Nicea being a suburb of Constantinople, Constantine's capital. He called the bishops of christianity to him. He personally presided over this council, and personally set the agenda... What he wanted and what he got was a unifying statement of belief that would unite all the different churches into one united catholic Church.. catholic meaning unified and being written into the creed. That statement generated at Nicea is today the Nicea creed which remains the most widely used christian profession of beliefs across Catholicism and much of the Protestant churches. I've even heard the Mormons use it.

Among Constantine's other innovations was the use of Christian relics. Body parts of martyrs or saints or even items which had come in contact with Christ himself. Constantine collected these totems and used them to court favor from christian leaders, or demonstrate his great knowledge / wisdom about christianity. This was the start of centuries of fascination with relics by the christian church which would only be curbed by Martin Luther's reforms.

Constantine was primarily motivated by the phenomenal wealth of the great temples to fill an imperial treasury bankrupted by civil war. Considering that his personal life never reflected any Christian virtues and that he remained a sadistic killer his entire life, he doesn't seem to have been changed much by his position as champion of the new state religion.

Even after his supposed "adoption" of Christianity the numismatic record shows that he continued to associate himself with the Sun God "Sol Invictus" (whom some claim to be a title of Mithras) and there is little credible evidence that Constantine was in any way serious about the new religion - the "church history", the alleged correspondence (which presents him as an obsessive expert on all kinds of ridiculously trivial ecclesiastical matters) and the ludicrous biography of the emperor - all published posthumously by Eusebius are transparent works of propaganda, almost comical in their hyperbole, and given Eusebius' own advice that lying to serve the interests of the church is acceptable and admirable, it is highly improbable that there is any truth to his works or to the unlikely tale of the emperor's deathbed conversion.

The truth is that he saw the astronomical wealth of the ancient temples as the easiest way to pay off and disband his restive armies and transform an empire in economic chaos into one of prosperity to enable him to begin the massive program of public works that culminated in the construction of Constantinople on the site of a minor Greek trading port, Byzantium (Byzantion).

As stated, he first de-criminalized it and converted much later if at all. When he stopped the persecution of Christians they had become so numerous as to be part of the standard Roman society. Removing them would be untenable for the survival of the Roman system. Also, they were responsible for looking after the many poor and destitute. Among the many cults growing and shrinking Christianity's stoicism and pragmatism meant that Constantine could not ignore it and could benefit by being seen to join it.

This question assumes that Constantine was a calculating politician who converted to Christianity for personal advantage and not out of pure holiness.

Your assumption is (half) wrong (and half right). As to the former (half wrong) part, most people living before the European Enlightenment were theists, and leaders, be they political or military, were usually no exception to the rule. As for the latter, see below.

What strategic value did Christianity hold for Constantine that made him convert to Christianity?

His continued rule as Roman Emperor, by trying to please or appease the specific deity he personally deemed responsible for his military victory over his political adversary. (Whether this is to be construed as an expression of deep religious fervor, or of pragmatic self-interest, is another question altogether, pertaining more to the realm of psychology, rather than history proper).

Coming from a polytheistic background, after having previously been raised in the paganism of his father, his acceptance of said deity was, at least at that moment in time, not necessarily a conversion to monotheism, especially since he personally was not properly introduced to (nor particularly acquainted with) it before, his mother Helen's Christianity notwithstanding.

Basically, shortly before fighting the decisive battle against his rival claimant to the throne, Maxentius, he (obviously) requested assistance from the gods (as almost any man of his times would have done under similar circumstances). He then noticed a certain starry formation, which, according to the superstitions of the time, he took as a divine response to his request (but he was a soldier, not an astronomer, by profession, and -even if- back then, in prescientific times, there was no meaningful difference between astronomy and astrology). Then, a certain Christian soldier, of which there were many in the Roman Imperial army, sensing the opportunity at hand, proposed a certain Christian interpretation of the perceived event, which he, desperate for some glimmer of hope (no pun intended) during the upcoming bloody ordeal, gladly accepted.

The main reason he was only baptized on his deathbed consists primarily of the fact that there was a certain pious belief going around in his time, based on a particular interpretation of a specific passage from Paul's Epistle to the Hebrews, that sins committed after baptism might be unforgivable. (Many fourth century Church Fathers, whose devotion and piety to their faith no one seriously questions, have also been baptized late in life, despite having been born to Christian parents, so there is no real reason to believe that Constantine's similar delay is based on any other reasons).

Admittedly there was a certain decline in the practice of infant Baptism during the fourth century. At that time even adults postponed their Christian initiation out of apprehension about future sins and fear of public penance, and many parents put off the Baptism of their children for the same reasons. But it must also be noted that Fathers and Doctors such as Basil, Gregory of Nyssa, Ambrose, John Chrysostom, Jerome and Augustine, [...] were themselves baptized as adults on account of this state of affairs [...].

— Source.

If we assume his mother Helena had no influence on his spiritual beliefs (big assumption), the major classical reason for conversion are military successes attributed to God's divine protection of Constantine, especially the Battle of the Milvian Bridge.

The gradual increase of Christians in the Roman empire can't be wholly dismissed either; as the young Christian Church could be useful source of taxation revenue and political legitimacy; even before being formally acknowledged.

The calculating politician answer: He gains reputation as a just and pious ruler (during his own lifetime and rule) and gains a group of fervent political advocates that offset any dissipated pagan factions of the time.

Put simply, early Christian missionaries ate away at native Roman religious groups until some Emperor was going to have to acknowledge them; since the other approach (of purging Christians) clearly wasn't working.