You see old-fashioned, I see classic. Isabelle is a variant of Elizabeth, as others have already pointed out, and it goes back hundreds and hundreds of years, just as Elizabeth does. Elizabeth in fact goes all the way back to the Bible. It's never been out of the top 100 names for girls as far back as records have been kept, I believe. Everybody knows how to spell it, everybody knows how to pronounce it, it's classless. It's a fantastic name.
Also, Elizabeth is just as likely to be shortened to Beth, Betty, Bette, Lizzie, Eliza. My impression is that Liz is far less commonly used these days.