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You have a beautiful name, which could sound strange in english and I understand that you wouldn't want it even slightly associated with lice. So many people go by nicknames and you shouldn't have a problem with that. I would stick with Liz, since you already like it and have been using it for years. All the other options are nice too, (except from Isis which is a really risky choice right now), but you will have to get used to them first.
My personal favourites from the ones you suggested are Ava and Sophie. I also loved Ivy from the names suggested, it s such a wonderful name, and I would add Eve and Zoe!
I hope all turned out well with your baby and you got to choose the name you like. That said, I think that your family tried to scare you into naming your child after the paternal grandfather. I am a greek, born and raised in Greece and I know for a fact, that although it is very common for first children to take the paternal grandparents' names, it is not uncommon to chose another name (most commonly the maternal grandparents' name, but some people chose a completely different name). Moreover, it is not true that our father's name is our middle name and a lot of people have two first names!! (none of them is considered a middle name, as they are equally important). My official name is Athanasia Ioanna V.....i and my father's name is Athanasios. I don't go around writing I am Athanasia Ioanna Athanasios V.....i!!!!!! Growing up, there were at least 10 children in my class who had 2 first names. Having said that, I understand that you wouldn't like George or Giorgos as a second name either and I don't blame you, it is too common. Georgia is even worse, as it sounds bad in greek and it means agriculture!!!! "Georgis-Τζώρτζης" and "Georgina-Τζωρτζίνα" sound better and if you try hard you will definately find a priest who will agree to saying the full name in church and signing the nickname, so you will not have to see the name ever again. Friends and family members equally have done this. Also, you should condsider Iris for your little girl, and say it derives from Georgia. I hope everything went well. (we certainly don't belong to our father and nobody asks us "who owns you?", I don't know where that came from).
I prefer Sophie, since the name derives from the greek "σοφία" or "sophia" and it has such a unique meaning, wisdom. Spelling it with an f, just seems to invalidate the name's history.