Cecco, Checco, Chicco, Ciccio (common in the South - even though it can sound as chunky/chubby - and there’s also “Ciccillo” in Naples), Franci (but it’s used also for the feminine Francesca, especially in the North), Cesco (rare).
@M_C_ to be honest, I don’t like pet names. Francesco is a good name as it is. Maybe you can use “Fra” just to make it shorter. Or just choose the option you like the most, it’s something personal, I guess.
I agree with Gino: there are differences between North, Center and South. Here in Tuscany we don't really use Checco/Ciccio and similar. We use Fra, France, Franci - older people could use Cecco o Cesco.
Apart from everything that has already been written and that is absolutely correct, I might add that "Frank" - as in English - could be also used (I have a friend named Francesco, and we call him Frank).