I would rectify why that hot doesn't have its neutral running with it. If its (either white wire) an old switch leg its absolutely the wrong color and I would proceed very carefully and inspect every inch of that house's wiring. More likely as mentioned its probably power to a switch and getting a neutral down the line. I don't see any reason why it shouldn't be nutted together (the ground wire also, definitely) But if it is power to a switch then its not connected at the other end either anyway. Making future additions unnecessarily complicated as all of your wires aren't hooked up like you think they would be. GFCI's don't like funny business on the neutral so keep that in mind.
Close that box and go about your life. Romex is pretty easy to run. While your at it and have your load center (breaker box) open, add a convenience outlet next to it so the next time you feel like cheating its right there for you.
-nuts should be spun so much that the wire winds around it self making a better psychical hold.
-use something bigger than #12, or you may have bad voltage drop depending on the distance.
-always run more capacity then you need at the moment. You will find something to plug in where that just isn't enough for you and be wishing you had.