Victor's Cafe, on W 52nd near Broadway, might be just what you're looking for. Very good Cuban cooking (rice and beans and fried plaintains are both great, by the way), family-friendly. They may be a bit on the pricey side for you... check out the entree prices on menupages.com, you can assess for yourself.
There's also a kind of Cuban/Mexican hybrid called Cafe Habana on Prince Street. Cheaper, still a great place for rice and beans. It's not a Cuban dish, I don't think, but the grilled corn there is very good.