How to save a state - make an arduino run a code only once/make aruduino bahave differently the next time he is on
As one of the comments said, you will need to use the EEPROM library. When you save a variable in your code (e.g. int test = 10;) the variable along with its value gets stored in the RAM. The EEPROM is a piece of harddisk on your arduino that will remember its value when you power off your arduino.
If you implement a power-off button, simply save the state (or in your case, the wheel size) in EEPROM whenever the user inputs it. The next time you power up, you can check if there is a previous value in EEPROM and continue with that value if it exists. Otherwise ask the user for their wheel size.
You might want to implement a reset-button as well, to reset whenever you change bike or wheels. Look up a code example of EEPROM, as there are a lot online.