If you do not have any special gameplay ability such as a gravity beam or something like that it all comes down to pushing buttons, switching levers, pushing crates and so on, simply because the way you can interact with the game world is limited.
In my game I have a level where you need to turn two somehow connected wheels (one moves when you turn the other one) in 45° degree turns to transmit electric current from one place to the other. You do this by jumping on one of the triggers below the wheels. The difficulty is that there is a flying enemy that chases you and the wheels turn rather slowly (and the correct order of turns is not obvious), so you sometimes have to get rid of him using your acrobatic moves. To complete the puzzle you need to pull down a metallic rail from the ceiling by grabbing on to it with the "rope" ability, you have to use your skill to avoid the enemy while hanging on the rail because the latter moves slowly, too. This is how I've tried to combine my jump and run gameplay and a puzzle mechanic.
Sometimes there are regular switch "puzzles" and sometimes rather not so obvious puzzles, but most if not all of them are realized by pressing/ pushing things.

I don't think you should try to avoid switches and similar/ known ways to interact with the game world, just try to make their effects interesting.
Maybe a few more ideas for you: Light and darkness (maybe put a light switch in a room and when the player switches off the light, he will be able to see a path/ note drawn with invisible ink, or the player can run over a puddle of invisible ink and make a path for himself that he can use to see in the dark), electricity, water, weight, air, ...