I think I like Amber because it's more distinctive than other jewel-themed names like Ruby and Jade. Of all the names I can think of in that category, I like Pearl the best.
Lilah is very popular on MN - maybe in real life too - but it sounds a little like 'liar' to me. It's a shame because I think it looks very pretty written down. Lilac is an interesting alternative but I'm not sure I'd be brave enough!
Based on the names you've posted, I think you ought to consider Sadie, Camilla (Milly), Ivy, Willa, Elodie, Isla, Eliza, Cecilia, Eleri, Ellery, Emma, Melia, Dinah, Delia, Cordelia, Blythe, Maia/Maya, Gaia, Iris, Cleo, and Camellia too.
This one might be a bit 'out there' but how about Elysia ('from Elysium') instead of Emilia? Bit different to Emilia, Alyssa, and Elissa and sounds similar to Emilia.
But I don't think any of the choices on your short list would be bad, really - just not to my taste (as they are not distinctive enough: too many Emilys, Amelias, Amelies and Emilias; too many Layla, Lyla, Leelas; and Amber might just be a touch 80s/90s but it's definitely my favourite).