Terpsichore - unusable. I think twerp and harpsichord.
Amphitrite - unusable. amphitheater
Mnemosyne - unusable. nobody would be able to pronounce or spell it
Nemesis - unusable. crazy.
Eurynome - unusable. you're a gnome.
Pepromene - hmm. It's weird but not completely crazy. Might be too much like pepperoni though.
Asteria - usable. I prefer Astoria, but Asteria is pretty as well. Unusual but nice.
Andromeda - usable. I like it. Andy or Romy would be cute nn's.
Honestly, I think only Andromeda is usable.
I automatically think Astoria when I look as Asteria as well.
Andromeda is by far the most usable here. And I really like it. I'd suggest Ariadne as well.
Nemesis is probably the only other one, besides Asteria, that people wouldn't struggle with, but would you honestly name your daughter something that means adversary? The mythological backstory isn't appealing either.
Terpsichore - it's pronounced terp-SI-kaw-ree. The muse of dance. She's one of my favourite muses, but I don't think it's usable as a first.
Amphitrite - it's pronounced am-fi-TRAI-tee. A pre greek sea Goddess. Too difficult, and not very pretty.
Asteria - lovely. Titan Goddess of nocturnal rites. I think it's a lovely name, and the most usable on your list. Asta, Ria, Teri.
Pepromene - personification of fate. Not very pretty, I wouldn't use it.
Mnemosyne - the mother of the muses. I think it's a pretty name, but probably as a middle as I assume a lot of people mispronounce it. Nemo.
Nemesis - the Goddess of divine retribution. I don't have a big problem with her, but it's not a name I would ever give to a child. Nemi.
Eurynome - quite pretty. The Goddess of water pastures and mother of the Graces.
Andromeda - Beautiful and haunting. An interesting story, and she married Perseus. And it's a constellation, and that always sways me. Andy, Meda, Mea, Romy.