One of my players recently acquired a protocol droid whose stats are meh at best is there a defined mechanic for upgrading the stats or skills? If not should I allow this and if so how?

Stats are easy enough just using the cybernetics in the book. There are some for a couple few skills as well.

How is the player wanting to upgrade it? Are they wanting to go to a shop and buy a better processor? Or are they a mechanic trying to do it themselves?

If they want to just purchase new upgrades for it then it wouldn't be too difficult to form something. For example, since it's a protocol droid they want to increase its negotiation or charm skill. The player could go to a droid parts store and purchase an upgraded linguistic module. Buy a new processor to increase knowledge skills.

If they want to modify it personally then folklore the mod rules for weapons. Programmers were able to change a droids personality or learning capabilities by tweaking the numbers in a sequence of its behavioral patterns.

Those are just a few things that occur to me as logical and rule based. I always fine the rule of thought as a GM to let things happen as long as it's reasonable, even if it may create annoying or more work for me. I'm about to create an economic system for galactic hunting and a skill that's applicable for training beasts because two of my PCs stories/ambitions call for it. Hope my input is helpful.