Here is another uncommon interpretation.

Assume that Daniel 10's reference to the third year of Cyrus' reign is not a reference to his reign over all Babylon, but the start of his reign many years before.

• 605 BC: Daniel carried off to Babylon
• 559 BC: Cyrus begins reign
• 556 BC: Third year of Cyrus' reign, angel delayed 21-days
• 535 BC: Daniel returns to Jerusalem after 70 years exile and dies
• 33 AD: Jesus crucified

Tradition holds that Daniel returned to Jerusalem according to prophecy after seventy years in Babylon and died that year.

Thus Daniel died 21 years after meeting with the angel.

Acknowledging that there is no year zero, the span from 556 BC to 33 AD is 588 years. Thus from Daniel's vision to the crucifixion is an even multiple of 21 years.

588 years = 28 x 21 years

Solomon in Ecclesiastes taught that there are twenty-eight times in certain prophetic cycles. If in this cycle the duration of the period for each time is 21 years, then:

• A time to be born: Begins in 556 BC
• A time to die: Begins in 535 BC
• A time for peace: Begins in 5 AD, Ends in 33 AD

This tells Daniel when he will die (535 AD, a time to die) and when the Prince of Peace will arrive (circa 5 AD, return from exile in Egypt) and depart (33 AD, end of a time for peace at the crucifixion, resurrection and ascension.)

What are common explanations for the angel being delayed for 21 days in Daniel 10:13?

This is not a common explanation, but I believe it is the correct explanation:

When you read the chapter and read in verse 1 that Daniel "understood the thing, and had understanding of the vision" (it is repeated for emphasis) you must wonder, like I have done many times in the past, "What on earth is there to understand? There seems nothing much here!" Then a few days ago I thought What if the 21 days delay is part of the message?

Just as Jeremiah was told how long the Babylonian Empire would last, 70 years, so Daniel was told how long the Persian Empire would last. The difference is that the Holy Spirit chose to keep the duration of the Persian Empire hidden in a vision: this vision is recounted in Daniel chapter 10. Daniel himself understood the vision (verse 1), but the Holy Spirit did not want him to explain it in the word of God.

The 21 days represent the 210 years starting with the Fall of Babylon 539 bc and terminating in the death of the last Persian Emperor Artaxerxes V, i.e. Bessus, at the hands of Alexander the Great in the summer of 329 BC.

Post script:

There may be another explanation instead of or in addition to the above.

Daniel had this vision "in the third year of Cyrus" (Dan 10:1). In his first year he issued a decree that the Temple in Jerusalem be rebuilt (Ezra 1:1-2).

[The "Cyrus Cylinder", a copy of which is in the entrance vestibule of the United Nations HQ, was issued for a different temple and is strongly suggestive he issued the same decree for all temples throughout his empire.]

The date the third year of Cyrus is dated from his conquest of Babylon which fell on the night of 12th Oct 539 BC. According to Babyonian reckoning his Accession Year was from Oct 539 to Nisan 538. His first year started Nisan 538, and his third year began Nisan 536. It was in this year that Daniel had the vision.

Why was Daniel "mourning three full weeks"? Maybe because he had heard that the temple had begun to be rebuilt, but had been brought to a halt by the enemies of the Jews. The delay of 21 days may be a declaration by God that the temple would not be finished for another 21 years, a year per day. The temple was finished on 3rd Adar in the 6th year of Darius I (Ezra 6:15), which according to Parker & Dubberstein's "Babylonian Chronology - 626 BC to AD 75" was 12th March 515 BC (Julian).

The Jews used inclusive counting. In this case it means that the remainder of Cyrus's third year after the time of the vision is counted as a whole year.

So the babylonian year containing 12th March 515 BC (when temple was finally completed) began in Nisan 516 BC which using Jewish inclusive counting was 21 years after Daniel's vision in the year starting Nisan 536 BC.

The delay of 21 days is God's way of telling Daniel the temple would not be completed for another 21 years, which Daniel understood, but the Holy Spirit did not direct Daniel to reveal it to the reader explicitly.

What are common explanations for the angel being delayed for 21 days in Daniel 10:13?

I believe common explanation miss something critical in this vision, so this is not a common explanation.

Daniel's mourning was in the third year of Cyrus. Cyrus conquered Babylon in Oct 539 BC. His Accession year was from October 539 to the end of Adar 538 BC. His first year started Nisan 538, and so his third year started 1st Nisan, 536 BC. Daniel mourned 3 weeks ending on 24th Nisan so it started 4th Nisan 536 BC. It is to be noted that Daniel mourned all through the festival of the Passover and the Week of Unleavened Bread.

The big problem in this period of Jewish history was in the rebuilding of the Temple at Jerusalem. Cyrus had authorised it, and the foundations were laid in his reign, but thereafter there were considerable hindrances, the enemies of the Jews causing trouble and delay. According to John Calvin "the Prince" is not a demonic angel but rather the man Cambyses II who hindered the work of rebuilding the Temple.

So John Calvin says:

We gather from this passage why the angel appeared to the Prophet in the third year of Cyrus. He says, he was then in the greatest sorrow; and what was the cause of it? At that period we know an interruption of the work of rebuilding the temple and city to have taken place. Cyrus was gone a distance away; he had set out for Asia Minor, and was carrying on war with the Scythians.

His son Cambyses was corrupted by his couriers, and forbade the Jews to proceed with the rebuilding of their city and temple. The freedom of the people might then seem in vain. For God had promised the Jews in glowing language a return to their country with their standards unfurled. Besides this, we know the splendid language of the prophets respecting the glory of the second temple (Isaiah 52:12; Haggai 2:9, and elsewhere.) When thus deprived of all opportunity of rebuilding their temple, what could the Jews think except that they had been deluded after returning to their country, and God had made a show of disappointing expectations which had turned out a mere laughing-stock and deception? This was the cause of the grief and anxiety which oppressed the holy Prophet. We now understand why he mentions the third year of Cyrus, as the circumstances of that period, even at this day, point out the reason of his abstinence from all delicacies.

The hindrances of the Jews' enemies seem to have been accentuated by Cambyses II himself because there was no progress on the Temple during his reign from 530 to 522. Finally the Temple construction restarted in the days of Darius I (Hystaspes) at the encouragements of the prophets Haggai, and Zechariah, and completed on 3rd Adar (the twelfth month) in the sixth year of Darius (Ezra 6:15), which was 12th March 515 BC.

Using Parker & Dubberstein's dates in "Babylonian Chronology, 626 BC to 75 AD", from the date of the beginning of Daniel's period of mourning, 3rd April 536 BC, to the completion of the Temple (Ezra 6:15), 12th March 515 BC, is 21 years minus less than a month. And using inclusive counting the year starting Nisan 516 BC was 21 years from the time of Daniel's mourning in 536 BC.

Each day of the 21 days of mourning thus represents a year until the completion of the rebuilding of the Temple.

It was God's revealed will that the Temple be rebuilt. Of course, no one, neither man nor demonic angel, can thwart the secret plan of God, so the thwarting and the delay were part of God's secret plan. But evil forces, though they think they thwart God's plans, actually help fulfil God's plans, and the timing of God's plans.

Apart from scholars involved in higher criticism, Christian commentators tend to fall into two broad groups in understanding the reason for the angel's delay. However, there is broad agreement that the author intended us to understand that the 'princes' in this passage were angels.

One group says that the 'prince of the kings of Persia' was an evil angel. James E. Smith (Daniel: A Christian Interpretation, page 319) says that the prince of the kingdom of Persia obviously must be a powerful angel, and that he must be satanic because he opposes the good angel.

Others say that the 'prince of the kings of Persia' was the guardian angel of Persia. The New American Bible footnote to Daniel 10:13 (5) says the later Judaism ascribed protecting angels to various groups of human society, often as little more than personifications. D. S. Russell (Daniel, page 199) agrees. He surmises that Persia's guardian angel resented the ministering angel's mission and the revelation he brought regarding the vindication of Israel. Russell says that according to contemporary Jewish belief, when a particular guardian angel gained ascendancy over his fellow-angels, the nation over which he had been appointed gained the ascendancy over its neighbours.