A good puzzle is one where the difficulty is just right from which the player will derive satisfaction from clearing it, feeling smart from having them use their brain even just a little. Show the goal/cheese/carrot on the stick and have them figure out how to get there. It's a lot like designing secrets, but with special attention and care.
Your adversary is the map, more so than the monsters in it.
Several common puzzle types in Doom:
Pattern puzzles:
- Finding discrepancies in patterns, memorizing them, or discovering them all on your own towards the path of success. Examples would be finding which of several torches is a different color from the rest, memorizing a sequence of teleporters, or hearing a lift activate when moving over a line.
Combat puzzles:
- How monsters will interact with the level geometry and actors themselves. Examples would be provoking infighting, calculating what monsters to alert in sequence, or just plain ol' slaughter but I wouldn't know anything about that because I don't play slaughter.
Navigational puzzles:
- Reaching your destination solely though the map geometry from leading a mouse to cheese, timed lifts or crusher passing, to platforming. Examples would be parkour secrets where riding along a ledge awards curiosity, finding which crates you have to climb up to reach your destination, or jumping down from on high to land somewhere you couldn't cross directly.
Play with sector height, brightness levels, thing placement, or even chains of linedef triggers. Anything you can think of to provide a sense of satisfactory noggin' joggin' depth in the map design of your level; and it's not like the bar of what's called a puzzle in Doom is all that high either!
Theoretically ambitious puzzles:
More ambitious puzzles are the kind that are deliberately vague and may require multiple attempts to reach a definite solution. These are harder to design with their deliberate cryptic nature but if done well are sure to be appreciated. Providing not just a goal, but a sudden obstacle will turn on a player's critical thinking switch. When requiring a set of actions to get from point A to point B, there should also be things that reverse those actions, which will require the player to think critically. For example, a switch could open a door into a new area, but if the player crosses a bridge, it shuts the door. Now the player has to think of a way to get through the door without using the bridge.
A different kind of puzzle I'd like to see done are narrative puzzles (that I would like to ask of people to make some of in the future) where the hints are derived from environmental storytelling or through information dumps of text the likes you would see in Doom 3's info terminals and PDA e-mails. Anyone should be able to figure these out with some sleuthing. Execution will be challenging but I think I'll be satisfied with just the joy of the craft itself.
It is even possible to utilize powerups such as the Light Amplification Goggles and Invulnerability sphere to play with how they change the lighting in a level. In fact the Radiation Suit can be used to read hidden messages written on the walls and whatnot. You can do this with palette/colormap changes.
Be a kind dungeon master:
It's okay to have obscure puzzles, the kind that people grill Eternal Doom for but I'm confident that such ambitions can be better designed through proper subtle hints towards the correct line of thinking in the map design itself or otherwise.
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If your puzzle is somewhat esoteric then make sure you lock the player inside of the puzzle room until they solve it. It'll be bad for everyone if the player wanders outside of the area not knowing what to do thinking the solution is elsewhere. Make sure there's a logic to it throughout because madman adventure game logic is hardly ever satisfying. I learned this lesson after making my DBP11: Lilywhite Lilith map. No one likes being lost or watching others be lost. It gets old and quickly!
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Make sure the player has an idea of what a switch did, that the player isn't allowed to wander wayward in search of a solution that doesn't exist, and there the player isn't overwhelmed by an abundance of teleporters and switches to confuse oneself with. If I see 20 switches and 8 teleporters all at once I'll just panic!
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Provide the player with a sense of progress. If the player does something that's related to reaching the goal, you can provide some health bonuses, but its better to unleash some monsters to kill to prevent the player from dying of boredom!
Thank you for reading:
Most of what I've written are my own words but it's still a mess of pilfered content. Additional credits go to @JadingTsunami for bringing the Radiation Suit trick to my attention and @40oz for some of his do's and don'ts I've also used as a checklist as seen in this thread.
AMOREUPHO.zip 
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