In Middle English, "y" was a regular variant of vocalic "i". For example, in the Canterbury Tales:
And, for to festne his hood under his chyn,
He hadde of gold ywroght a curious pyn;
A love-knotte in the gretter ende ther was.
His heed was balled, that shoon as any glas,
And eek his face, as it hadde been enoynt.
He was a lord ful fat and in good poynt
(lines 195 - 200)
Here, "y" is used for "i" where we'd expect the "oi" digraph; but it's also used for "i" itself (lines 195 - 196). Similarly there's a variant "ay" of "ai". Again from the Tales:
At mortal batailles hadde he been fiftene,
And foughten for oure feith at Tramyssene
In lystes thries, and ay slayn his foo.
This ilke worthy knyght hadde been also
Somtyme with the lord of Palatye
Agayn another hethen in Turkye.
(lines 61 - 66)
It's this "ay" for "ai" substitution which gives us the Middle English word "taylor" (which as you correctly deduce is in fact the word "tailor"), and the Modern English forename and surname "Taylor". (As a surname, this was an occupational name - i.e. given to a person who was a tailor by trade.)
After spelling got regularized in the 17th or so century (don't quote me on that :-) ), the common nouns went back to the "i" spelling, but names often stayed with the "y" spelling.