The six solfège syllables ut, re, mi, fa, sol, and la were, as mentioned in another answer, derived from a hymn, so there's no reason to seek functional similarity between, for example, fa and la.
Ut was later replaced in Italy with do. (France still uses ut.) The guy who came up with this, a fellow by the name of Doni, had a bit of a hard sell, and one of the points he made was that it is the first syllable of dominus, meaning lord, but I don't see any suggestion that it was thought to be advantageous to have the same syllable there as in sol, though there is certainly such a case to be made. (It doesn't seem that he stressed the fact that it is also the first syllable of his own name, though he does seem to have had this in mind.)
At some point in the eighteenth century, si was added, which became ti in the English-speaking world. In this case, I think it is pretty clear that the vowel i was chosen because of the very similar melodic function of the third and seventh degrees of the major scale.
While it is true, as noted in another answer, that fa and ti are both found in the very important dominant seventh chord, there are other contexts in which ti is more closely related to mi than to fa. In particular, mi and ti have similar melodic function because they are both a half step below their upper neighbor in the scale. Before the invention of si, an ascending C-major scale would have been solmized ut, re, mi, fa, sol, la, mi, fa.
Indeed, it is this property of the i vowel representing a melodic (upward) leading tone that led to the choice of the same vowel to indicate upward chromatic alteration in di, ri, fi, etc.
