Lydia and Jessica are both great, although I am not a fan of Ida...
Other thoughts - Rosanna, Rosalind, Louisa, Frances, Francesca, Adeline, Adelaide, Adela (Ida alternatives), Miranda, Serena, Cassia, Caroline, Marianne, Emmeline, Emma, Julia, Juliet, Lucinda, Hannah, Harriet, Cecily, Cecilia, Felicity (and the currently v popular Charlotte, Beatrice, Emily, Anna)
Rose is nice but it’s a cliche middle name, so might be better as the last of 3 names or a variation like Rosemary.