Congratulations on your new little girl.  She's adorable! My take on the name is 

 only becuase I know a guy called Izzy who is a real pain.
Your little girl is too pretty not to have a soft sounding feminine girlie name. 

 To me she looks like a Cali. (Cali-Gal) Or even Sunshine (Sonnie)  It's just my take on this since you ask!