How do I keep a player in the game when their character dies, without resurrecting that character?
Don't kill the players darlings
First, always make sure that the players are okay with losing their characters. Before running the game, explain that there might be character death and that they should create their characters accordingly.
I seldom run games where character death is a thing, but when I do I make sure the players know about it and are prepared for it before we start.
Bring a backup
Have a few backup characters in tow. Let the players create them so that they're invested in them and let them be mostly passive during play. This allows you to have new characters at-the-ready.
In a game I'm preparing now, the players will each create one "officer" of sorts and one "second in command". If their character dies, their replacement steps in. This also gives a reason for the secondary characters to be more passive.
One problem with this approach is party balance. The death of a character may not be a problem for the players, but the party looses in overall strength. This could, of course, also be an interesting challenge for them...
The chance encounter
A wild adventurer appears! Adventurer has joined your party!
Yes, it's a common target for ridicule and parody, but sometimes it just works. If the adventure is a bit cliché and campy anyway, this is a perfect fit. And a bit if sillyness is better than benching a player, right? Still, this is often not the best option so only use it when appropriate.
A more plausible reason for someone to be in the dungeon alone is to be the last surviving member of a previous, less successful, party of adventurers. Sure, the player will have to find some new gear, but that shouldn't be too hard. Hey, that other person who just died seems to have the same size as the newbie!
I've also used variations of this to introduce new players to an adventure or campaign.
The chance necromancer
A neutral necromancer lives in the dungeon. Who knew? She can resurrect the fallen character for a small fee. Or perhaps she's just imprisoned as well and will accept freedom as payment.
This option has to be done carefully. An undead PC can work fine as long as the player has fun with the character. Don't make the zombie too zombie-esque, let it be able to speak and retain intelligence. To make it feel a bit more like a proper death and not an upgrade, let there be a restriction to the resurrection so that the zombie can never leave the dungeon without losing its intelligence.
The reformed minion
After falling into a deadly trap, the characters are rescued by a minion of the dungeon. Before letting them out, the minion explains that he wants to change his evil ways and join the party. This can backfire if the other players decide not to trust the former minion, but it can also be a nice way to introduce a character with an instant backstory.
I've used variants of this to introduce new players and as long as the other players play nice it's a great way of fleshing out a group with some new blood.
The unreformed general
Give the player a major character in the dungeon. Perhaps even the main villain. Or simply invent a commander. Let the player look at the full map, plan a defence and command troops. This can become really engaging and rewarding for the player.
The downside is again that the party will become unbalanced. Although it might also help the rest of the party work together to keep each other alive in order to not gain another maniacal enemy...
The talking artifact
Want a weird twist to the adventure? Let the player take on the part of a talking artifact, such as an enchanted sword or shield. Try to make sure the thing is too big to shove into a backpack but also too valuable to leave behind.
I've never tried it myself, but the more I think about it, the more I want to.