I really like
Ida, too!
To me, this is a classy but still very fresh choice for a girl.
I used to like it even more but it has gotten very popular over here. I live in
Europe, by the way. To me, it is a Nordic name. So I also pronounce it [EE-dah]. I don't really like the English pronunciation.
Iva is alright, but I definitely prefer
Ida.
Iva reminds me of
Ivan which I loathe.
Ida is sweet and short, therefore I would add a longer, maybe -e ending name to it. I don't like to see it in the middle name spot because it is so short.
Suggestions:
Ida LiliIda CecilieIda CharlotteIda HenrietteIda KarolineIda MathildeIda PaulineIda RoseIda Sofie
Of the combos you suggested I like
Ida Josephine, but would prefer
Ida JosefineIda Marian, though I find
Marian a little masculine sounding