Ida... You're right, many things are very clunky with this one. I guess I'd pair it with something very airy, probably longish too, probably a bit flowery or princessy.
Something like:
Ida Felicity Ida CelandineIda CalantheIda CamilleIda CelesteIda JasmineIda LavenderIda SaraiIda SerenIda VashtiIda Flower ? Maybe not.
Ida Garland - this isn't soft, and kind of heavy, but I kind of like it anyway
Ida Leilani Ida Lorelai - ha, no
Ida MaylisIda Thalia
Sorry, it got very flowery there towards the end. I kind of wish you were pronouncing it "eye-da". Then I could suggest:
Ida Leigh (idly)
Ida Lynn (idling)