In the First Apology (chapter 67) Justin the Martyr wrote that the community normally celebrated the Eucharist in the 'day of the sun'. The theological motivations given by the author are two: first because in that day God started the Creation, and secondly that Jesus died on the vigil of the 'day of Saturn' and resurrected on the 'day of the sun'.

The author put first the creation and then the resurrection, but we know that it is part of his style put last what logically should be put first (hysteron-proteron), so we may suppose that the 'resurrection argument' is the first motive of the celebration in that day, and the 'creation argument' is secundary.

Justin dedicated his Apolgy to Antoninus Pius, that reigned from 138 to 161, so we can affirm that Eucharist was for sure on Sunday in first half of second century.

In an ancient liturgical books, the Apostolic Costitutions (book 7,36) we can also find this prayer (that some authors suppose being an ancient eucharistic prayer) that can help us to identify the motivation between the substitution of the Saturday with the Sunday: it state that the Saturday is the memorial of the many things that God did in the Old Testament, but the Sunday is the memorial of Christ, His Incarnation, Death and Resurrection, and this is greater than the prodigies of the Old Testament.

The long eucharistic prayer of Apostolic Costitutions 8,12,4-51, ancient for his subordinationist and angelomorphic theology of the Son and the Spirit, also start the praise with the Creationd and end that part with the Death and the Resurrection, that are reconnected with the Last Supper.

IF we want to synthetize we can say that the Sunday was rapidly identified with the day in which Christ resurrected. Because in the Ancient Church the Eucharist was the memorial/sacremental ripresentation (tzikkaron) of the Death and Resurrection, rapidly was chosen as the perfect day for the weekly celebration. the theme of 'Day of the start of the Creation' came to reinforce this practice.

And then there was the powerful driver of the contrapposition with the Sabbath on Saturday, in which the Jews commemorate the Old Alliance and the Liberation from the Egypt. The Christians believed that Jesus stipulated the perfect Alliance, which superated the Old.

sources:

- The Anaphoral genesis of the insititution narrative in the light of the anaphora of Addai and Mari. Acts of the internation Liturgy congress Rome 25-26 october 2011; pages 439-440 (for ApCost 7,36)

- In unum corpus. Trattato mistagogico sull'Eucarestia page 260 (for Justin)

- you can find the original text in greek for ApCost 8,12,4-51 here and here for the text of Justin