I have an "Italian honey" fig. They are excellent when dead ripe. But, this year many (mostly, high up in the tree) have a tough, brown area on the end. What causes this? 

Rain.

Eden, I thought of that, as a posibility, but this (second) crop wasn't even on the tree when we last had rain.

I get that on some fig varieties when it gets very hot and dry here in the summertime. Some varieties seem more susceptible to this than others. Extra water seems to somewhat help, but not totally.

They heard about the rain?

Sunburn perhaps?  Happens sometimes on the citrus here if there is not enough leaf cover.