I used a Pfllueger Medalist 1492 for backpacking for years. It held enough line, had a good, strong click & pawl and wasn't bothered by being dropped against the occasional rock. 40 yrs. old now and still going strong. If you'd like something a little more modern but still inexpensive, the Okuma Sierra (about size 4/5) would work. I've even seen it on sale for under $20.00, but usually it's more like $30/35. Maybe not quite as tough as the Pflueger, but still plenty stout. It has a disc drag, too, not that one is needed for most backpacking situations. Good luck!
Troutcatcher6, you are the drag for that reel, how awesome is it that you are in absolute control of what that fish does. I just started fishing a dragless and I'm not that good at it but I'm liking it a lot.
A Medalist is a good reel, but they're built a bit heavy for backpacking. A Berkley 554 or 556 is very light, and long out of production, but you might find one on Ebay for cheap. Or any cheap superlight reel from stamped aluminum should be good for backpacking. A backpacking reel for trout fishing doesn't need to be all that durable.
There are lighter options for a backpacking set up, but I'd put a J.W. Young Pridex or Beaudex in the same camp as a Pfllueger. Classic English clicker. They can be had on ebay for $40-$45 and they're pretty stout.