Well, it's not the best solution, but it might be the lightest. I used to take a few feet of a worn fly line (taper isn't really an issue in this case), a few feet of tippet, and a Sucrets box with a few flies in it. (OK, I have been known to take a few hooks and split shot in it too.)
If a fishing opportunity arose, I'd cut off a "whippy" green sorta straight limb, tie some line and leader on it, and have it it. Of course, this was in the southeastern US, where small streams and lots of vegetation abound. But it was sort of on-the-fly (Lord, I apologize) Tenkara fishing.
Normally, though, if I was backpacking and was pretty sure I'd be fishing, I'd carry a multi-piece rod (four or more pieces), in the lightest tubing material I could find, and lash it onto my pack (or stuff it inside). I'd take the lightest reel I had (no shopping for a new super light reel), and a lightweight box with 2-3 dozen dries, nymphs and streamers in it. Didn't take nippers, fly flotant, net, etc.
Mostly, though, I take way too much stuff, because I'm so used to, well, fishing with my loaded vest. It's equipped for most any fly fishing situation, and, probably, a night in the woods, minor first aid treatment, maybe even minor car repair.