Thamyris was a Thracian bard or minstrel, the son of Philammon and the nymph Argiope (and allegedly the first man to love a boy - Hyacinthus). He even dared to challenge the Muses to a 'music and poetry' contest. Predictably, they won, and exercising their right to deprive the loser of the contest of anything they wished, took away his sight as well as his poetic gift.
The root of his name appears to be thama, 'together in crowds', 'close', 'thick'. This is difficult to apply to one person, so perhaps it is better to consider the other sense of the word, which is 'often', 'frequent'.
This could refer to his frequent singing and lyre-playing: perhaps indeed his name is a contraction of something like 'Thamalyris', so that it contains not just the thama element but also lyra, 'lyre'.