[Verginius Rufus]

You're referring to words in modern Greek. I can't find anything similar in ancient Greek in the large Liddell & Scott Greek-English Lexicon. They list a verb byo (beta-upsilon-omega) meaning "to stuff full of" with a few related words like the adverb byzen (beta-upsilon-zeta-eta-nu) meaning "close-pressed" (Thucydides uses it of the Spartans lining up their ships close together at 4.8.7). Other than that there are a few rare words known from lexicographers, but nothing related to breasts.

Sometimes when a new group comes in and takes over a local name from the previous inhabitants, they modify the name to make it meaningful in their own language. For example, La Jolla, California, supposedly meaning "the pearl" in Spanish, may have been an Indian name which the Spanish colonists modified. So the Greek colonists could have modified a native name in the case of Byzantion.

[Nikas777]

byzen (beta-upsilon-zeta-eta-nu) meaning “close-pressed”

Same thing - the origin of the word to suckle comes from the baby being close pressed against the breast to suckle.

[Nikas777]

PS: I am not discounting the Thracian origin either but the Greeks have plenty of examples of men and nymphs copulating and finding cities from that union.