Have a good curve
This is basic but worth mentioning. Pacing must be kept carefully with new mechanics to ensure the player has time to grasp the mechanic and grow comfortable with it before being asked to use it in more complicated ways. In games that offer players a lot of different abilities - think Banjo-Kazooie - these are often unlocked one at a time, with more and more available later in the game.
|Otherwise, people might get confused by something like this.|
If you've played a 3d Zelda game, this probably looks familiar:
|Take a moment to appreciate how ordinary this looks for how weird it is.|
|ok maybe I play too much nintendo|
Late in Twilight Princess, Link gets a staff capable of controlling big statues and moving them around. They have to be a specific kind of statue (statues with a hole in the chest are our "cue"), and they can't jump, so they're bounded near their starting point by terrain. This tool is only useful for controlling these statues; it has no effect, of any kind, on anything else, and no other tool can be used to interact with these statues. In this case, the item is situationally-useful; it is specifically restricted to only one kind of use. Combined with cues, figuring out which tool to use is no longer part of the puzzle; the puzzle is figuring out how to use each tool effectively. This is also how games like Braid divide up types of puzzles; some mechanics, like the shadow copy of Tim or the time-slowing ring, only appear in certain areas. The game allows for variety by having these mechanics, but only applies them in certain situations, so the player isn't confused about which mechanic to use.
At first glance the idea of redundant mechanics may seem like a design flaw, and in some cases it is. Sticking with Zelda, though, think of how enemies work in Skyward Sword. For the most part, they require more sophistication than simply smashing them with your sword, but often there are several different ways to combat them. For example, a Beamos can be defeated by using horizontal slashes to cut it down to your size before stabbing it in the eye - but it can also be killed with a well-timed (and well-aimed) arrow to the eye. When you encounter several of these enemies over the course of the game, it adds to the fun to be able to mix up how you deal with them. Moreover, the arrow strategy is a little easier and safer, but you get the bow long after encountering your first Beamos, and when you do it's a nice upgrade for dealing with this kind of enemy. Redundancy is most common for combat purposes; you have a few different abilities, and one might be more efficient than the others for taking out your foe, but it doesn't really matter which one you use. There's no hard and fast limit to the places where you can offer players multiple solutions to a single problem. You could do this in a puzzle game (though at the risk of reducing the difficulty of the puzzles), and it's common to see it in RPGs, where it might be possible either to fight or talk your way out of a dangerous situation. Redundancy either requires careful balance of all the available options or demands that newer, better options render old ones obsolete, and so must be used carefully. When implemented well, however, it can give players a sense of accomplishment when they acquire a new tool that renders old challenges much easier, or give them a sense of ownership of the story when they can solve problems on terms of their choosing.
There are a lot of great games that have few mechanics and never give the players new abilities; simplicity is a beautiful thing, and some games would have their elegance diminished by overcomplicating the mechanics. But variety can be fun, and it's neat as a player to see that you have a lot of options or that you gain new skills over time. A good way to summarize the points I talked about might be to ask of any mechanic:
When was this mechanic introduced?
What lets me know I should use this ability/mechanic?
Are there places where I'm forced to use this ability, or where this ability does not work?
Can this ability complement, be used instead of, or replace altogether another ability?
Hopefully the above will give you something to think on next time you get a cool new toy in a game, or the next time you're thinking about implementing one in your own game.