A Mistake to Learn From
I picked up a router base from Rockler designed for centering on board edges. It's simple: two brass pins screw into the base and straddle the edge of the board. You gently twist the router, keeping the pins against the board's surface as you cut. It works nicely. But here's what I found out by mistake (mine), so I hope it helps others.
I had five mortises to cut on a three-foot long board's edge. The first four went smoothly. The last one was about two inches from the end and that's where I learned that, as you cut the mortise, if a pin passes a point where the board's surface ends, you'll lose its support and twist the router right out the other side of the mortise. Which I did, much to my chagrin.
If you have to cut a mortise at the end of a board and you won't have support for the pins throughout the length of the mortise, clamp a piece of wood the same depth against the end, forming a butt joint, and then cut the mortise. In other words use another piece to "extend" the piece you're working on.
Fortunately for me the resulting gap in the side of the mortise is quite small and easily repaired.