What are must-have items in an emergency kit for a multi-day hiking trip?
Obviously, what you decide to pack depends on your trip and the environment, but these are the staples I always bring along:
Navigation: You always need to be able to figure out where you are and which way you need to head to get out. In my opinion, a map without topography isn't a map (unless you're on very flat land).
Warmth & Shelter: You need to be able to get warm and protected from the elements. I always bring a sleeping bag if I know it'll get cold at night, even on single-day hikes where I don't plan on sleeping in the backcountry. God forbid you get lost and have to spend the night, it's worth the extra few pounds for peace of mind.
Rescue & Signaling: If you fall and break both your legs or get your arm pinned under a rock, you need to be able to make a commotion and attract attention.
Food & Water: This isn't just so you can stand on a mountain eating a Nature Valley bar like they do in the commercials. If you need to survive for a few days, you'll appreciate the sustenance.
General Survival: If you have to make shelter, conduct first aid or perform any other basic camp tasks, you'll probably need this stuff anyways.
Edit: As Phil mentioned, you should always bring a cell phone. Keep it in a waterproof bag, and turn it off so you're not disturbed. But having one in a pinch is the difference between being able to summon help for yourself and sitting around hoping someone is out there looking for you.
There are different types of being stranded; there's "lost the trail an hour in" stranded, "lost the trail twenty miles in" stranded, and "broke a leg" stranded, just to name a few.
Considering your question
Your emergency kit should be gear that never leaves your person. If I get stranded away from my gear, I have this kit. People have died from going to take a leak in the middle of the night and getting lost. Your general hiking equipment is your "gear," moreso than your "emergency kit".
Why "never leaves your person"? You are at primary risk of two types of problems. For simplicity we will assume the worst case in both, that you have lost your group.
If you won't have your emergency gear with you, then it's not an emergency kit. With that in mind, I build my kit for three things
Any kit is nothing without knowledge. You need to know how to use all this stuff, how to make your own shelter, and preferably all the local edibles and how to find them. Before you go out on a day hike, I recommend you practice with this gear to make sure you can do the minimal of:
This stuff never leaves my emergency kit. For any of it that I also use regularly in camp, I have a second one. My emergency kit is for emergencies only and is separate from the rest of my gear. It has its own pouch that I keep with me (not with my pack).
I hike the Appalachians. It gets cold, but not Canada cold, and it's not desert. This would likely need to be tweaked for extreme cold or heat.
Hartley (and Phil) pretty much have it covered, but I'd add a couple of things.
Also, it's a little over the top, but we take emergency cards that you fill out with vital information like the nature of the emergency, how many people in the party, an emergency contact for your party, your location etc, and send someone with the card to get help.
For a multi-day hike obviously you'll have a tent which can be very useful in emergencies.
Just additional two cents, but don't underestimate the visibility of a green laser pointer at night. In survival situations and with light fog, a green laser pointer shooting up in the sky will make your position extremely clear.
For the exact same reason, don't use it with airplanes around. They hate that, and you will be found and prosecuted.
The other answers focus on general survival kits, but don't give much coverage to first aid kits.
Here's what you need in a first aid kit, in order of importance:
That's it. Everything else you might need can be improvised from regular hiking equipment and the ten essentials.
My list may not sound as sexy as some of the others, but it's been tested in the field. I'm a former EMT and climbing guide, and I've faced multiple life-or-death medical emergencies in the wilderness.