Most of those solutions measure movement, not force. If you really want to measure braking force, I'd go with:
- Direct measurement of the wire tension. Similar to this geometry.
but instead of a tactile switch, I'd use a strain gauge force sensor to measure the force on the middle wheel in that diagram. That way, the middle wheel hardly moves and the travel of the brake lever won't increase significantly.
People! Surely the following should have been addressed before now!
The goal is to trigger an 'action' as soon as the brakes are used, not when the effect of using them becomes apparent (or not).
It is worthy of a thread in its own right! A deep discussion of fundamental Heisenberg issues!!
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But to turn to something useful...
Do you need to know how hard/ how much braking? Or merely that braking has been initiated??
If the latter will do....
A weatherproof (your environment IS challenging!) pushbutton switch kludged onto the thing you squeeze to apply the brakes, and held shut (pressed up against some part of frame) when you don't squeeze. When you squeeze, pressure on the button released, "braking" detected!
No, let's not stray from the topic please. Quantum mechanics gives me a severe migraine.
We're dealing with macroscopic observables here, no need for collapsing wave-functions / PDFs and CDFs to measure stuff on a bike.
On the brake handle, between your hand and the grip, put something like this...
The harder you squeeze, the more braking.
Or was something like that already suggested?
Sort of 
And I have my doubts about how long this piece of plastic would last under my grip. Linear potentiometer it is for now.
A little update.
I very crudely attached the sliding potentiometer to the bike and used one of my old LED projects to show how strongly I pull the lever. The principle seems sound.
Looks good, minor remark, from esthetical point of view I would one LED array move clockwise and the other counterclockwise so it becomes symmetrical.
Another option is to make the pattern "logarithmic" -> 1 red -> 2 red -> 4 red -> 8 red - would be nice symmetric too.
That said, Well done!!
Commencing symmetrization!
Now I need to look into weather-proofing this thing. Last winter I used a plastic bag. That works, but it is not what I'm after. Ideally I'd make a permanent project and get some sort of transparent coating and a battery box under the saddle.
looks better - more professional - imho.
You can also add a indication you want to go left or right by blinking the appropiate side.
PS, silly effects vid was good too !