I think Izzy was the name of my childhood friend's iguana, come to think of it.
But yes, it's a cute nickname, and I like how Izi and Isidore have that little extra something without sounding wacky or misguided ("This is my daughter, Annavanessica! With a Q."). Who knows; if no Isidores turn up on reality television in the next ten or twenty years, I may report back with news of so-named spawn!
I have a friend here in Ljubljana who named his son Izidor – the Slovene version of Isidore; he is now 14 and everyone calls him Izi (pronounced like the English word "easy"), which he thinks is pretty cool.