I can't tell too much from the pics, but it looks Greek, rather than Roman or Byzantine, and bronze. There's a female head (or perhaps Apollo) on the obverse, and something I can't quite make out on the reverse, perhaps a bird in a tree, with Greek text underneath it - I can see a letter phi.
Bodrum is, of course, the modern Turkish name for the ancient Greek city of Halikarnassos. In ancient Greek times, bronze coins rarely travelled far from where they were struck. So the logical place to start searching is through the coins of Halikarnassos
; if we can't find it there, we start working out way out from there.
Don't say "infinitely" when you mean "very"; otherwise, you'll have no word left when you want to talk about something really infinite. - C. S. Lewis