So, what's your favorite type of puzzles in video games? Do you like to move objects from A to B? Perhaps grab an item C to ram it in slot Z? Or find a note containing a password to input in mcguffin Y?

I'm asking this because of the game we're making and we're looking for a way to make it more... game-y.

So, puzzles. What are the best ones?

I just finished Portal 2 yesterday.
Loved the experience of physics, out of the box thinking and the story/atmosphere.

Anything, so long as it's not timed.

This, pretty much.

It's hard to say anything more without knowing moe about the game. A geneal advice would be to make the puzzles feel like it makes sense for them to be... wherever they are. What I mean is, avoid things like safes of suitcases locked with some ridiculous, no way this would ever actually exists puzzle-lock or heavy machinery operated by chess puzzles etc. It totally breaks immersion and reminds the player that it's a game, and that what happend here is exactly the developers trying to make it more game-y. And it sucks.

The best puzzles in games are from puzzle games. I know that sounds like flippant remark but most games do not have, as a part of their base gameplay, a system that lends itself to proper puzzlemaking. A lot of what gets called a 'puzzle' in many games is not really a puzzle. What does get called a puzzle in many games is more often a variation of the find-a-thing sort of stuff that you seem to want to add to your game. I don't think of that as a puzzle, just a different way of finding a key to unlock a door.

Portal is a good example of a game that has base mechanics that lend itself to proper puzzlemaking. Most FPS are the opposite with 'puzzles' commonly limited to find-a-thing stuff. What kind of gameplay mechanics does your game have? With a decent game mechanic the puzzles you can include will be better than the common find-a-thing puzzle.

Yeah, timed puzzles are horrible...

A large part of me wants to agree.
But there is also a small part of me, trying to argue that some puzzles make no sense unless they're timed.

I really liked Dishonored 1&2 safe codes and granny rags quests - they are optional and really well built into a gameplay - without this artificial feeling.

In general - I like puzzles that involve exploration, attention to details

Blast to the past, but I loved those chest riddles in Betrayal at Krondor.

And I hate the "sliding tile" puzzles.

I also dislike puzzles where you have to plan dozens of moves ahead to get to the solution. Though that might just be because I suck at planning ahead.

One thing to keep in mind - unless you're making an actual abstract puzzle game, it's good to have the puzzles realistically integrated into the story.

Portal 1 & 2 are top notch.

I'm also found of the light/dark puzzles in Closure (https://www.gog.com/wishlist/games/closure *wink* )
Really simple concept but really cool and atmospheric game.

Also, Zelda fanboy here so navigational/environmental puzzles or just the usual and formulaic "sliding block into button" puzzle in between stabbing monsters and I'm fine with. xD

Lufia 2 has my favorite game puzzles. There are lots of different types it uses, but there's a special place in my heart for the yellow and red block puzzles.

Physics puzzles are also really great, if only because of how amazing it feels to find unintended ways of finishing them.

Shiren the Wanderer has Fay's Puzzles, which I enjoyed immensely. The game is a roguelike, but there is a special place you can enter in the starting town and play through some puzzles (50 in total). Those puzzles use the normal game mechanics (and don't let you bring in XP or items), but instead of standard procedurally generated roguelike fare, you go through hand-made dungeon levels, and have to figure out how to complete them using just what the game gives you. What's interesting is that the puzzles sometimes require you to do things most players wouldn't normally think of, like intentionally trigger traps that knock you back long distances, intentionally equipping cursed items because they're cursed (not because it's the only option or because of the game hiding the cursed status of the item), throwing money (yes, literally), and I believe there's one that requires you to steal from a shop.

Another interesting example is the geo-panel puzzles found in the Disgaea series. Basically, the game is an SRPG, but there are geo-symbols which produce an effect that affects every geo-panel of the color it's on, and destroying them can cause the colors to change. A rather interesting idea, with the small issue that it isn't colorblind friendly. (Accessibility issues should be considered when adding puzzles to a game; otherwise, such a puzzle might make the game unplayable for some players; Chrono Trigger requiring multiple simultaneous button presses at one point is one example of an accessibility fail.)

Ditto
And there was always a key to the answer if you just knew where to look.

Dungeons & Dragons Online has a large Mastermind type puzzle in its first Stormreaver raid. They have also employed large, coloured, turnable wheels with symbols, and the keys are colour/symbol combinations placed elsewhere in the dungeon.

As previously mentioned, the context, setting, and mechanics of your game and the specific area in which you want to place a puzzle will determine whether a given puzzle will make sense or seem out of place.