3rd white wire in electrical outlet with different power source?
Any idea why this outlet would be wired this way?


Because someone before you didn't follow basic electrical practices. Was this wire connected to anything when you opened the box?  Was this a standard duplex outlet? The only situation I would think of is if the outlet was wired with one half to one white wire, and one half to the other white and sharing the black - but that would be horrible.

Label the that wire with something obvious and the number of the odd breaker and put a wire nut on it.  Replace your outlet and secure the outlet (but don't fully button up the box yet)

Turn on the power to the outlet breakers and check to see everything works.  Then turn on power to the other line, and test everything in the house.  If something isn't working, you'll have to start investigating the rats nest of wire in the walls/ceiling/floor and try to determine how it should have been done originally.  

Hopefully, everything works with it capped and you can secure the box and go on your happy way.  When you rip out the walls for some reason, plan to have someone rewire as needed.  Who knows what else is messed up.