I know there is already the Tinkerer talent that allows you to increase an item's HP. But the Jury Rigged talent can supposedly increase any 1 characteristic of an item, and I can't find any mention of this rule excluding HP. If Jury Rigged can upgrade HP, would that mean you could use Tinkerer and Jury Rigged on the same item for +2 HP?

If there’s a specific talent for it (like Tinkerer), then I would say that Jury Rigged can’t be used to replicate the same functionality. Otherwise, there would be no purpose to the other talent.

That just seems like an obvious abuse.

You can find the full rules for Jury Rigged on Page 138; it pretty much stipulates that you can do the following;

Increase the damage of a weapon by one, decrease the advantage cost on it's critical, or any single other effect by one. (Weapon)

And then it goes onto saying similar things for Armour, though it does rather specifically say what it can do.

Given that a Hardpoint is not really an effect per say, I'd go ahead and say that no, it cannot do that.