Since there is no projection on the end of the button then there must have been a plunger part, also probably spring loaded as it cannot make contact with the shaft in operation. It would almost certainly be round so if you exposed the arbor of the router you could see what size hole the plunger would fit into. You might be able to see into the cavity or try working blind with different size drill bit shanks until you find a fit but then you still have the issue with how to keep it out of the way when operating it.
Someone I consider a master woodworker once told me that a master woodworker is not someone who never makes mistakes. He is someone who is able to cover them up so that no one can tell.