This riddle isn't too difficult if you brainstorm out possible solutions using the letters given. After all, the "letters that stay" include only one vowel. That greatly limits the number of possible solutions. There is also the clue about it being the name of a Saint, which is a great way to be able to check your answer to be certain it's correct.
The "seven letters" part is a little confusing because it implies that the solution will have seven letters in it, but with only one repeated letter that gives a maximum of five. I chose to ignore this as probably referring to the original seven letters in Clarice.
With a little deductive work, we're left with the name Clara. St. Clara (also known as Clare of Assisi, Clair, and other variations) was an Italian Saint and the first woman to have created a monastic "rule of life."