Years ago when I first got my 10/22 one of the holes in the top of the receiver stripped out.  I recently decided to replace the cheap scope mount with dovetail rings, but before I can I need to fix the hole in the rear of the receiver.  Has anyone used a heli-coil or other method for repairing the hole?  I have tap and die sets but am hesitant to tap one size up.

I have never used a helicoil, but I have re-drilled and tapped these to a larger size.  I am using a Volquartsen scope mount, which I also had to modify slightly.  Take care to grind the screws down so they are flush with the inside of the receiver after installed so the bolt doesn't hang up on them.  

In my mind, larger screws means more surface area and more strength.  The receiver material is not all that thick, so a helicoil may not have enough to grab on to.  I suppose one could always try some acraglass to fill in the extra space but probably not as good as just re-drilling and tapping.

That's what I did with good results. Had to enlarge the hole in the rail and as others have said , grind the screw from the inside of the receiver.

There really isn't much meat for a helicoil to get a grip when you consider it is soft aluminum, the larger screws are a better repair.

Having installed (and removed) 1000's
of Helicoils in my life, I say:

DO NOT "Helicoil" this hole!

Upgrade to a larger fine thread, or if you're "good enough"
find a place next to hole and drill & tap there, and drill
and (counterbore) mount to match.

(If you're using a long mount with room to do this)

You'll hide the buggered-up hole and avoid having
to re-tap the rest of the holes to match.

If you're using an upgraded mount that requires larger mounting holes,
upgrading all of the holes would be cool, too.

Brownells sells  oversized 6-48 tap and screws. They also sell a Locktite product called Thread Restore.You can also go to 8-32 but you'll need to redrill the countersink in the rail.

I went this route a few years ago when I ran into a similar problem.  
One of the filler screws was stuck hard and absolutely would not come out.
Wound up drilling out the filler screw and ordering a screw and tap kit from Brownells.
Problem solved.