My general process is starting with a property of one of the mechanics or an interaction between multiple mechanics that I find interesting. Once I have that, I think about some interesting outcomes of that property/interaction, specifically looking for ones that differ from what you could normally do if that object/group of objects wasn’t present. Then I just need to make a layout where that outcome needs to happen in order for the puzzle to be solved. If nothing comes to mind, I usually consider adding additional mechanics to make these differences more meaningful (the webs are actually a good example of this). This is probably way too vague, so here is a quick example from the game:
Snow: Vault
In this case, the property I started with was that snow can be destroyed instead of needing to be pushed out of the way. One outcome of this is that snow trapped against the wall isn't an obstacle like a typical crate would be. There are a lot of potential ways to make use of this property, but I thought the most interesting one was using the snow to push two wood blocks simultaneously without getting them stuck to each other. From there, it was mainly about finding the right button placements. The hardest part of designing this level was actually just making sure the player couldn't accidently stumble into the mechanics explored in areas 4 and 5.Stephen’s Sausage Roll pretty heavily shaped my taste in puzzle mechanics, it has a similar style where most of the mechanics are introduced right at the start and later puzzles just force you to interact with them in new ways.