When did the number three begin to get fixed, though? There's an interesting web page with some useful background here:
Concerning the Magi and their names
This is from a site called The Hymns and Carols of Christmas by Douglas D. Anderson. The page quotes a certain Otfried Lieberknecht who cites Metzger:
A good study to check in cases like this is Bruce M. Metzger, "Names for the Nameless in the New Testament: A Study in the Growth of Christian Tradition", in: Kyriakon. Festschrift Johannes Quasten, ed. Patrick Granfield & Josef A. Jungmann, vol. I, Münster: Aschendorff, 1970, p.79-99. Metzger has a whole chapter (p.78-85) on "The Names of the Wise Men", which includes a rich bibliography of earlier studies of this topic (p.79 n.3).
According to Metzger, the earliest source giving names to the magi are the anonymous _Excerpta Latina Barbari_, the Latin redaction of a Greek chronicle which seems to have originated in Alexandria during the first or second half of the 6th century: "In his diebus sub Augusto kalendas Ianuarias magi obtulerunt ei mundera et adoraverunt eum: magi autem vocabantur Bithisarea Melchior Gathaspa" (cit. p.80).