If someone with a 1 in a stat got a Gruesome Injury I'd just make sure it didn't go to that stat.

I remember looking around and not finding rules that specify if anything happens to a character whose characteristic drops to 0. We do have an example of actors without characteristics to go off of: Droid brains. Droid brains have skill ranks with a characteristic of 0, so they only roll green dice. The same should work for a PC. If it's a skill that they have no ranks in and the characteristic is 0, I'd let them flip a destiny point to "upgrade" to one green die, or they can see what boost dice they can add up and go with those. They're already suffering from a nasty critical injury, I don't see a need to pile additional penalties on top of that.

And, how would you handle a Gruesome Injury to someone with a 1 in the characteristic? Ignore the effect?

Yes,... and maybe no...

How about treating the Gruesome Injury like a Horrific Injury until it is healed, then swap back to 1; best of both worlds. Would that be to your liking?

The Gruesome Injury critical entry specifically states that it cannot reduce a characteristic to less than 1. I would assume that this also applies to Horrific Injury, since the rules make no allowances for characteristics being reduced to 0.

If you really needed to, though, you could always let the characteristic temporarily drop to 0. This would have no apparent effects beyond forcing the character to roll only his skill ranks when using skills associated with that characteristic, or only boost dice if he has no ranks in the skill. Personally I wouldn't bother, since he already has a critical injury that must be healed or give him +10 to subsequent critical injury rolls, which is enough of a penalty in itself.

And, how would you handle a Gruesome Injury to someone with a 1 in the characteristic? Ignore the effect?

Yes,... and maybe no...

How about treating the Gruesome Injury like a Horrific Injury until it is healed, then swap back to 1; best of both worlds. Would that be to your liking?

I do like that a lot, but I feel like there should be a permanent effect when healed if the characteristic was a 1 to start. Maybe a permanent setback die to all rolls associated with that characteristic that cannot be reduced by talents or maybe a permanent increased difficulty (not upgraded to red) to all checks associated with that characteristic. If the character takes dedication to raise the characteristic to a 2, then this "permanent" effect will go away.

Thoughts?

We had a player get a gruesome injury to brawn, which he had a 1 in. Right before hand we had just found a stash of power armor (with the +1 brawn), so we put him in that so he was able to move. Unfortunately the next encounter (a trap) had a temporary EMP effect, which shut down his power armor (as well as my droid character completely). He was unable to move the entire fight. He was a force user though and could use force move for stuff. The dungeon was all pre-scripted by the GM, but he did not expect us to do it in that order or for the player to have a brawn of 0. Happy coincidence.

TLDR: we played it as brawn 0 means you cannot move normally.

I do like that a lot, but I feel like there should be a permanent effect when healed if the characteristic was a 1 to start. Maybe a permanent setback die to all rolls associated with that characteristic that cannot be reduced by talents or maybe a permanent increased difficulty (not upgraded to red) to all checks associated with that characteristic. If the character takes dedication to raise the characteristic to a 2, then this "permanent" effect will go away.

Thoughts?

My thought is that having a characteristic at 1 is its own permanent penalty. If it doesn't feel that way, then it would be better to push harder on that in other parts of the game rather than piling additional bad effects on a rare critical injury. For one, you run the risk of penalty feeling disproportionate because it's coming all at once, and for another, if your goal is to push the players to round out their characters's stats, it's unlikely to accomplish that, as the most likely take away is "don't get a gruesome injury," not, "invest in boosting a characteristic to 2 in case you get that 1% chance of it being even worse."

As for how I'd run it, I'd be inclined to say "No effect," because what that probably actually means is "No effect... beyond another Critical Hit that will be a pain to remove and make further criticals more dangerous." And if they're already getting crit rolls that high, an extra +10 is probably going to have them really sweating. After all, Gruesome Injury +10 is Bleeding Out, and no one wants to go there. They'll probably thank their lucky stars it's not worse than having to worry that the next hit will be fatal.