A wiring diagram for the ignition switch would be best.
Do you mean to replace the ignition entirely, or just add a starter button? If the latter, then all you need to do is identify the starter wire and use a normally-open switch to put 12v on it when closed. Key goes in the ignition, key gets turned to "run", button is used to start. Key is turned off to kill the engine.
If you mean to replace the ignition entirely, your situation gets more complicated, since you need both start, stop, and run positions. Maybe a three-way switch that springs to center, with up being "start" and down being "stop" and center being "run". But then you've always got the mower in "run" and accessories can run down the battery. So maybe a "stop/run" toggle switch and a "start" button would be best?