This is the spelling of the name used by Mrs. Clinton, whose fame has temporarily taken this name off the table for babies. A pity, really, as it's got so much going for it: the popular, rhythmic three-syllable structure, the fact that it's strong but light, proper but jaunty, with an irresistible meaning -- having the same root as hilarious. The Italian Ilaria may be one way to make the name fresh.
Traditionally, Hilary was the form used for males and Hillary for females, but since there are so few boy Hilarys these days, that tradition has fallen by the wayside. A few other modern notables have laid claim to the name: musicians Hilary du Pre and Hilary Hahn, and actresses Hilary Swank and Hilary Duff.