My mom has a tree that produces bad tasting dried up figs. I believe that it grew wild. My question is "can I graft a branch from a healthy tree and get great figs"?  

It is probably a caprifig. I have never heard of grafting onto a caprifig but I guess it would work. The figs on the graft might split sometimes because of over-pollination if you have the wasps in the caprifigs there.

You can graft any other fig onto it you'd like, and it will produce figs, but it won't change the fruit produced on the non-grafted branches.

Thank You Brent and Jason,

I don't know how big the tree is but I would just cleft graft the entire main trunk if possible and top the whole tree with a desirable variety. With a good healthy rootbase the grAfts will grow very quickly.

Thanks Cal, I will try that cleft grafting.