AideenMongan's list "I Girls" of 4 great name ideas: Iphigenia - Imperia!

I Girls

This is a list of girl names that begin with the letter I so it is

In mythology, Iphigenia was sacrificed by her father, Agamemnon -- a difficult legacy to pass on to a daughter, and only one reason the name is hardly ever used. 

Iphigenia

Strong and sharp but with a pretty, poetic feel about it, Iseult appears in the world of Arthurian legend. Also written as Yseult, Ysolt, and Isolde, the name might have Germanic roots, blending is and hilt to give it the cool, wintery meaning of "ice battle".
 
The figure of Iseult was first mentioned in a 12th century French text, in which she accidentally takes a love potion and falls in love with Tristan, the nephew of the king she was meant to marry. Since then, the forbidden lovers have appeared in literature, art, and film countless times.
 
Pronounced as i-SOOLT or i-ZOOLT among English speakers, EE-salt in Ireland, and EE-ZUU in France, international forms include the Italian Isotta and the Slavic Izolda.

Iseult

Sister of Antigone and daughter of Oedipus in Greek mythology, Ismene's name is unlikely to make it in modern times because of its teasability factor.  Ismay or Esme would be better bets.

Ismene

Rather imperious as a baby name, but has an interesting history. Imperia is the name of an obscure French saint, also known as Impère and Impérie, also borne by the famous Italian courtesan Imperia Cognati. Honoré de Balzac later used it in his short story La belle Impéria (1832), where it belongs to a fictional courtesan. A similar name, Bel-imperia, was used by Elizabethan dramatist Thomas Kyd for a character in his play The Spanish Tragedy.