I'm from a small town in Appalachia. My folks were among the first settlers in America. The old timers claimed that the name "Ida" when put with "May", in place of Mary was to hide the line of Mary queen of Scots. Daniel Boone's people hid the name in "Ida-May" or Mae. The Irish pronounced it this way. Ida-May meant I'm the Mary. And so the Scots queen was hidden in every little girl called Ida. It was to throw the British off the trail, so that the true Mary, and the true queen would never be found. There's an old blue grass song called "Ida-Red". "Ida-Red" stands for "Ida-May". In the song, Ida-Red becomes Ida-blue and Ida-Green because this is what the Scots had to do to hide Mary's line in America. They changed the name, and hid it in so many families, stories, and songs, the true queen will never be found. We just use it these days to poke fun at the English!
Not introduced by the Normans, but before the conquest Ida was a man's name in England. In fact an Ida was king of Bernicia.In compound names it occurs in Idaburh, Idaberga, Idhild (f.) and Idesbeald, Idgerius (Idgar) and Idwine. The root survives in Icelandic iðja (work, activity), OE idig, idge, busy.Idisburh however may represent OE ides, OS idis, OHG itis woman, matron (in OE a poetic word applied to women of mythic or legendary stature).
Another possible theory for the origin of this name is connected to the namedays listed here. In the ancient Roman calendar the Idae marked the 15th or the 13th day of every month. So every Ida can choose her own name day or celebrate every month. I personally always forget to do it.In Bulgarian there is a verb that sounds the same way and means "I am coming" and in Spanish there is a noun meaning "going, departure".