Let's see. French-Canuck, Irish, Catholic. Form used in Shakespeare. Contains "A".
From Hobbs' helpful list, Francisca and Margaret seem to fit best. Of the two, Margaret has the most variants...the most French being Marguerite. Similar Gaelic names are Marge, Madge, etc.
But Margaret isn't an iconic Shakespearian character, and that connection isn't likely to be made for most people....unless Bri is misdirecting us...
I've thought myself into a corner. I need one more clue. Till then, I guess Margaret.
edit: crap, didn't think of male names. List, Hobbs?