First off, as a fanatical fly fisherman, I'd be the first person to encourage you to take up the sport but I actually think a fly rod is a poor choice if you are planning on adding a fishing rod for general purpose use on backpacking trips. I only take my fly rod when my backpacking trip is specifically a fly fishing trip (ie. I'm packing in to a lake or river and plan on spending 8-10 hrs per day fishing).
For all other trips I take a small ultra-light graphite spinning rod with a small reel. My rod is a two piece 5.5 foot rod with a small spinning reel and weighs in at less than 11 oz. My additional gear will weigh less than 5 oz. bringing the total weight to 1 lb.
Now I will explain why the spinning rod.
My terminal tackle consists of all or some of the following depending on the regulations. To start with the line should be very light (4 lbs. is good) as it allows me to cast farther and is required to support the rest of my gear. I take flies and small bait hooks (#12), some small power bait eggs if legal, and some light weights (1/8th of an ounce) or split shot. I also take a small twist on bobber (rubber tube type that allows for easy depth adjustment). This gear allows me to fish flies on the surface using the bobber, chironimids below the surface with the bobber and nymphs along the bottom behind a small weight. This will work in rivers or lakes. If the flies don't work and it's legal you can always switch to power bait on the bottom with the small hooks... which works almost anywhere (however fishing like this should not be done unless you plan to eat your catch... catch and release doesn't work well with the bait option). I tend to shy away from the power bait option but when fishing is really bad I will try it.
The reason I do this is that I like to keep things light and if you take your fly rod you will have to decide on whether or not you need some kind of wading shoes or waders to get to the fish. You may also want to take multiple spools (sink tip, full sink, floating) which all add weight (and cost). You may also have to deal with the problem of packing a long rod unless you specifically buy a pack rod. The problem I have found with fly rods on backpacking trips is that access to water without my waders or a float tube is often a problem, especially when I’ve never been to an area before (I hate to carry the extra weight and then not get to use it!). You need a decent amount of back cast room and the places I hike to are often tough to fish even with a spinning rod.
For the fly fishing purists that feel a need to comment... I've been fly fishing almost exclusively for 35 years, am a fly tier and a rod builder and have heard it all before!