How to simulate waking the dragon?

Arkham horror uses an impending doom card for each boss monster, and has various ways to trigger increases and decreases in doom. When the number gets down to zero, the monster appears.

This could Easily be re-themed into wakefulness or whatever word you use to describe it, with ways in which his sleep can be disturbed by building the cage.

This direction might add another strategical dimension.

I think it could be more fun if the player who wakes the dragon loses the game. That could introduce some interesting decisions. For example, if one mage adds a particular big piece of the cage, it could make a lot of noise, possibly waking up the dragon. This way players have to balance risk (waking up the dragon, losing the game) vs reward (building the cage, winning the game).

How about an alertness score?
Arkham Horrors mechanic for the doom and terror tracks were mentioned, so how about having a track that measures how deep the dragon is sleeping.
Make a noise, remove a counter.
Each quiet turn add a counter as the dragon drifts deeper asleep.
This would create a dillema for the player; "do I act and risk waking the dragon, or keep quiet and risk the other player getting ahead"
You could combine this with a dice roll mechanic to make it more likely for the dragon to wake if there are more/less counters, for example. EG:
There are 3 counters, so to wake would require 3 1's to be rolled.
You then make a noise and remove a counter. At the end of the turn you now only need to roll 2 1's, making it more likely that it will wake.

I´d say an alertness score is a good idea to make it less random.
Yut you can use something to counter the awakening instead. Give every-player a sleep-scroll or something like this he can use once per game not to wake tha dragon on snake eyes.

Maybe it´s better not to use snake eyes, but something that is more likely later in the game.
Maybe roll 3 dice, and if the total is lower than the number of bars on your side, you wake the dragon.

Binary Dice ....

This is a system we have for a Fantasy strategy game that we will never publish, because of its proprietary nature and dependency on electronic media.

They are tokens that represent elements of a given gaming environment. Each having a positive aligned side and a negative aligned side. The dice are thrown and any one event or all events for anyone turn or round of play may be applied to the role, depending on the rules that govern the reading.

This sounds difficult...but it is not, As in a game like yours you may have:
1) One alignment token
2) Two player tokens
3) One dragon Tokens
4) Two cage tokens
giving you six random +/- then the role is ruled. with "If Then" statements.
like:
"If" the alignment token is Black (-) "then" only Black tokens are read.
"If" a players token is on top of 1 or more other tokens "then all tokens are read as that players points.

It is a 3 part system to control random events with a unique Mechanic. were the 1) elements represented 2) random events are ruled. 3) the reading rules, all are able to be manipulated by the designer.