What is a good resistor value for my circuit?
Operate the relay from the signal/voltage that powers the brake lights. 
This gives 12V when operated and open circuit (= 0V) when off (OR ground when operated and open circuit when off in some less usual situations). 

The 12V power line from the battery will remain at about 12V at all times under all usual loads except starting. The starter motor MAY load the 12V line down to 10V or less, but nothing else will.