I went to a 'posh' girls' school from 5-18 in the 70s/80s. The girls were mostly called Rachel, Sarah, Anna, Victoria, Rebecca, Claire/Clare, Julia, Katherine, Sophie, Emma, Caroline and Elizabeth. So all very, very standard names.
The girls at my DD's school (also "posh") are called things like Flo, Mollie, Millie, Alice, Matilda, Olivia, Izzie, Charlotte and Sophie. So, again, pretty standard.
There are a fair few names that would be regarded as a bit try-hard.