When is the first documented case of Christians praying to the dead saints?

The first case of prayer to the departed saints is documented in the divine liturgy. In the divine liturgy of St. James the Just the priest pray,

[G]rant that our offering may be acceptable, sanctified by the Holy Spirit, as a propitiation for our transgressions and the errors of the people; and for the rest of the souls that have fallen asleep aforetime.

The divine liturgy of St. James the Just, XXVI.

For us and for the rest of the souls that have fallen asleep. Even prior to the Marian dogma at the Third Ecumenical Council of Ephesus, Theotokos is mentioned in the prayer,

Hail, Mary, highly favoured: the Lord is with You; blessed are you among women, and blessed the fruit of your womb, for you bore the Saviour of our souls.

The divine liturgy of St. James the Just, XXXV.

The priest's prayer also include those in the joy of paradise,

Remember, O Lord God, the spirits and all flesh, of whom we have made mention, and of whom we have not made mention, who are of the true faith, from righteous Abel unto this day: unto them do You give rest there in the land of the living, in Your kingdom, in the joy of paradise, in the bosom of Abraham, and of Isaac, and of Jacob, our holy fathers; whence pain, and grief, and lamentation have fled: there the light of Your countenance looks upon them, and enlightens them for ever.

The divine liturgy of St. James the Just, XXXVI.

After the Apostles wrote their Gospels and Epistles, in the New Testaments prayer through the departed saints can be found in 2 Timothy 1:16-18 (St. Paul prayed for Onesiphorus) and in Revelation 5:8 (Saints in Heaven receive our prayers). These passages could be read in isolation to how that texts were read and understood historically. Catholic and Orthodox read the Scripture not as a lexicographic text open to interpretation but as a liturgical hymn understood by the Church universally through all ages. This is evident in our divine liturgy:

That the practice of praying for the dead has descended from Apostolic times is evident also from the Liturgies of the Church. A Liturgy is the established formulary of public worship, containing the authorized prayers of the Church. The Missal, or Mass-book, for instance, which you see on our altars, contains a portion of the Liturgy of the Catholic Church. The principal Liturgies are the Liturgy of St. James the Apostle, who founded the Church of Jerusalem; the Liturgy of St Mark the Evangelist, founder of the Church of Alexandria; and the Liturgy of St. Peter, who established the Church in Rome. These Liturgies are called after the Apostles who compiled them. There are, besides, the Liturgies of St. Chrysostom and St. Basil, which are chiefly based on the model of that of St. James.

Now, all these Liturgies, without exception, have prayers for the dead, and their providential preservation serves as another triumphant vindication of the venerable antiquity of this Catholic doctrine.

The Faith of Our Fathers, XVI, Purgatory and Prayers for the Dead.

This is a historical claim with a historical basis. Prayer for the dead and prayer to the departed saints are related. One is assumed in the other. In the divine liturgy both are included.

Most Christian groups accept at least four Ecumenical Councils to be in line with the Scripture. Throughout these four councils we can find:

1. St. Athanasius of Alexandria wrote a book on the prayer to saints and angels in the Life of St. Anthony.

O noble Virgin, truly you are greater than any other greatness. For who is your equal in greatness, O dwelling place of God the Word? To whom among all creatures shall I compare you, O Virgin? You are greater than them all. O [Ark of the New] Covenant, clothed with purity instead of gold! You are the Ark in which is found the golden vessel containing the true manna, that is, the flesh in which divinity resides. Should I compare you to the fertile earth and its fruits? You surpass them, ... If I say that heaven is exalted, yet it does not equal you, ... If I say that the angels and archangels are great—but you are greater than them all, ... If we say that the cherubim are great, you are greater than they, ... If we say that the serphim are great, you are greater than them all, ....

Athanasius of Alexandria, Prayer to Theotokos.

2. St. Basil of Caesarea wrote his divine liturgy with intercession for the living and the dead extensively.

Unite all of us who partake of this one bread and cup to one another in the communion of the one Holy Spirit. Grant that none of us partake of the holy Body and Blood of your Christ for judgment or condemnation; rather, grant that we may find mercy and grace together with all the saints that have been pleasing to You throughout all time: with our fore-fathers, fathers, patriarchs, prophets, apostles, preachers, evangelists, martyrs, confessors, teachers, and with all the righteous made perfect in the faith, especially with our ever-holy, ever-pure, ever-blessed and glorious Lady, the Birth-giver of God and ever-Virgin Mary.

(The people now sing a Hymn to the Most Holy Theotokos.)

With the holy prophet, forerunner and Baptist, John, the holy, glorious and praiseworthy apostles, Saint N., whose memory we celebrate today, and all your saints, through whose prayers visit us, O God. Remember also, O Lord, the souls of your departed, all those who have fallen asleep in the hope of resurrection unto eternal life. O our God, we pray for the forgiveness and the repose of the souls of your departed servants, Nn., in a place of light, where there is no sorrow nor mourning. Grant them rest where the light of your face shines.

Basil of Caesarea, Anaphora Prayer.

3. St. Cyril of Alexandria venerated the shrine of St. John the Apostle and closed the Third Ecumenical Council of Ephesus with a devotion to Theotokos.

Hail Mary, Mother of God, Crowned Treasure of all the Universe, Star without Decline, Crown of Virginity, Sceptre of the True Faith, Indestructible Temple, Dwelling of the Incommensurable, Mother and Virgin. We salute you because you are called Blessed in the Holy Gospel and you come in the name of the Lord. We salute you, Mother of God, because you contained in your virginal womb what Heaven could not contain. Through you, in whom Heaven rejoices, the Holy Trinity is glorified and worshipped in every land.

Cyril of Alexandria, Litany of Praise to Theotokos.

4. Pope St. Leo the Great approved the Chalcedonian Fathers' veneration of St. Euphemia and her miracle at her shrine in Chalcedon.

For it was God who worked, and the triumphant Euphemia who crowned the meeting as for a bridal , and who, taking our definition of the Faith as her own confession, presented it to her Bridegroom by our most religious Emperor and Christ-loving Empress, appeasing all the tumult of opponents and establishing our confession of the Truth as acceptable to Him, and with hand and tongue setting her seal to the votes of us all in proclamation thereof. These are the things we have done, with you present in the spirit and known to approve of us as brethren, and all but visible to us through the wisdom of your representatives.

Pope Leo I, Letter 98, From the Synod of Chalcedon to Leo, Veneration of St. Euphemia of Chalcedon.

These four historical facts are suffice to show what the Church historically believe. While one can't pick and choose what to believe from Charles Taze Russell or Joseph Smith Jr., without ceasing from becoming a faithful Jehovah Witness or Mormon. Some Christian groups could say that they were committing idolatry distancing themselves from their heresy while accepting their theology and councils' decisions on Trinity and Christology as long as they consider it according to their reading of Scripture. Catholic and Orthodox consider that to accept Nicaea, Constantinople, Ephesus, and Chalcedon historically is to accept those councils as how they were accepted historically.

Some Christian groups disagree with this view and consider that those four councils were fallible and subject to their reading of Scripture because they judge that the Church historically has been corrupted earlier prior to Nicaea gradually. Catholic and Orthodox argue that there is no principal difference to distinguish Jehovah Witnesses' theology and Mormonism from Protestantism because they deny what Christianity historically believed while favoring their own lexicographical reading of Scripture as the norm. More to that both argue it also assume that around fourth century the Church was corrupted significantly and the gates of Hell prevailed making Christ's promise null and void. Either sides do have compelling arguments accordingly. Because each depends upon what constitute historical Christianity? Some Christian groups consider the norm for historic Christianity is the New Testament, the rest of Church History is subject to Scripture. While Catholic and Orthodox accept the infallibility of the Ecumenical Councils and judge what constitute historic Christianity accordingly. But all sides agree that this is what the Church historically believed and practiced since the late 2nd century.

Saint Jerome (347-420 AD) spoke of the practice with approval, saying:

"If the Apostles and Martyrs, while still in the body, can pray for others, at a time when they must still be anxious for themselves, how much more after their crowns, victories, and triumphs are won!"

His wording indicates that it was a common practice at the time: so there is probably no record of a 'first case' going that far back.

The earliest may not be easy to find, but I'll put a marker down.

St Ephraim the Syrian, who died in 373, prayed:

Ye victorious martyrs who endured torments gladly for the sake of the God and Saviour; ye who have boldness of speech towards the Lord Himself; ye saints, intercede for us who are timid and sinful men, full of sloth, that the grace of Christ may come upon us, and enlighten the hearts of all of us that so we may love him.

— Encomium in Martyres

Here he is clearly invoking the prayers of martyrs (who have died, rather by definition).