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I am designing a dungeon crawler/skirmish game after seeing The Silver Tower and getting inspired. Lots of events in these can occur which could use an element of chance or skill to complete e.g. trying to open a chest, avoiding a trap, jumping a gap etc.
Now, in a GW game these would usually be solved with a dice roll and possibly a modifier e.g. roll 4+ to open a door. However, there is no skill involved in this and its also a little dull. So i would love ideas on how else these events can be solved that are fun and have some element of skill. Ideas i have so far include dice challenges (e.g. roll 2 6s' in 10 seconds to complete), a maths puzzle (!) and a finite resource system (e.g. jumping a gap requires 2 action points, but then you may be less able to fight).
Thanks!

Take a look at the board game, Space Cadets.

However, you will find this concept goes over very poorly if your target audience is similar to the one for Silver Tower. Space Cadets also got old quickly.

One way around this might be to expand your 'action point' idea to cover everything, but remove the dice element; thus, every 'action' automatically succeeds (for both players and GM) but you can only do so much. This means players need to think about what they're doing a lot, but also can be content that what they choose will work. This means you won't get 'wasted' actions as people spend AP then whiff attacks or the like, This might seem overly simple, but it could force some interesting decision-making as you can only get so much done. It also means you know that any enemy attacks ect will hit and damage, so you can't rely on them rolling low.

Alternatively, you could go halfway with this; you can spend 1AP to jump a gap on a 4+, or spend 2AP to automatically make the jump. This means players can avoid the element of chance if they need to, but also places a cost on that (namely, being able to do less in a turn). You could apply this to all kinds of things, attacking enemies (1AP for a regular die roll attack, 2 for an automatic hit, 3 for double damage). This again adds some more choice, and forces players to think about their use of AP; yes, they can absolutely guarantee killing that monster for 3AP, but it'll leave them able to do very little else that turn, which in turn prompts teamwork and such.

I advise you to consider a card mechanic, instead of a dice one.
Cards aren't only substitutes for dice, but a whole new universe: narrative, visual, managerial...

It is something I toyed with a lot and have seen a few "hybrid" boardgame systems ("ameritrash"/ "amerithrash"+ Euro) attempt it.

Mind you it is not in dungeon crawler format, but the mechanical ideas and results are the same and can be applied to dungeon crawlers.

It depends how much of a mind puzzle you want your game to be, the more it is, the less things must be secret.

The virtue of randomisers be it dice, cards, spinners, flipping a coin, drawing from a bag, cube towers and other ideas that escape me is that they give a chance for encounters to go in "legendary" ways leaving stories behind of abysmal defeat and victories against all odds, a purely mathematical model will make players calculating and not taking the risks characteristic to the genre.

The other thing mentioned is indeed the dexterity focused game system were the players dexterity is their skill, these systems have their pace and a few "dungeon crawling" themed boadrgame exist but they are in my opinion not that widely accessible, they usually also need less delicate pieces depending on the dexterity methods used, for example flicking.

Spinners are the worst, because they combine complexity and non-randomness to what should be simple randomness. If you want a spinner, just roll a die.

Flipping the dice or playing rock paper scissors is also a bad idea and probably not the worse used in a game, but I was just listing randomisers that I know that they have been used in the past.

Do you remember any other?

Yup. RPS is semi-random the first time you play a stranger. After that, someone's gonna get destroyed if they're playing seriously...