Tire pressure change when standing still....you should see an LSB flip with a 6 bit converter, as an example
Nice, event i think this one will be difficult to fulfill the budget requirement as it will need a wireless transmission, still it is interesting - which sensor do you have in mind?
Use L3GD20 ST MEMS motion sensor, it is 3-axis digital output gyroscope 3D sensor - the price is 2$.
&
LSM303DLHC, ST MEMS system-in-package featuring a 3D digital linear acceleration sensor and a 3D digital magnetic sensor - the price is near 2$ too.
Another nice idea, will gyroscope has enough sensitivity for this? I have MPU-9250 somewhere in here - may try it.
I still have several months for this, any other tough please speak

-ichan