Olivia - yes, it's popular but there are so many girls names that I don't think she'd be surrounded by multiple other Olivias all the time. I only know 4 under 8 years old.
Livia - love it. Much prettier than Olivia imo. I've only ever met one. I did mishear it for Olivia but only needed correcting once.
Anna - like it. Classic and well known but not overly popular.
Nina - not keen on this. Not sure why. It's ok.
Mila - quite like it. Can see that yes, it might date. But don't know if that really matters. You can usually tell roughly how old someone is anyway.
Ideas a bit like Olivia and Anna but less popular if that bothers you:
Astrid, April, Bronte, Bryony, Clarissa, Claire, Diana, Eliza, Edith, Fiona, Frances, Giselle, Harriet, Hazel, Imogen, Iona, Juliet, Joanna, Kathryn, Leah, Laura, Miranda, Melissa, Natasha, Natalie, Ottilie, Opal, Phoebe, Philippa, Rosa, Robyn, Sadie, Sylvia, Tamara, Tegan, Verity, Vanessa, Zara
Or more like Livia and Mila:
Aria, Aliya, Briar, Cassia, Carissa, Dana, Etta, Enya, Freya, Fenella, Greta, Gloria, Hana, Ida, India, Juniper, Kyra, Kayla, Lyra, Lena, Mika, Mala, Nia, Nyla, Orla, Piper, Ramona, Rowena, Saskia, Stella, Veda, Willa, Xenia, Zia