My first thought is a dowel, glued in place. This will stiffen it some, but leave the crack. And epoxy can be on the brittle side.
So next, I thought, why not use a metal dowel? It adds more stiffness, more strength, but you still have the crack.
Another idea was a piece of all thread rod. Screw it right into the hole. This will give a great deal of holding power, preventing the crack from widening. But still you have that crack to fill.
Finally, I realized the perfect solution. McFeeleys sells it, in the form of a LONG screw, that is threaded only on the bottom half. Pre-drill a hole to get it started, and countersink it, so you can put a plug in afterwards to hide it completely. Then use the threads to draw the crack together as the screw tightens down. Add some glue (or epoxy if you prefer) into the crack before this operation, and the board will stay solid forever.
You can buy these screws in various lengths as needed.