When the tenon is broken off, you must rebuild the end of the tenon. Hold the piece in a padded vise to avoid splitting the wood when drilling and chopping out the waste. Then cut a piece of wood to splice into the old wood, using the original mortise to size the width. Clean up the drill holes by paring the holes with a sharp chisel until you have a good fit with the insert piece. Make sure the grain is the same orientation, then glue the insert in.
Round tenons broken at the shoulder present a problem.
The best way to repair these are to cut off the tenon end below the shoulder at an angle of 30 degrees or less. Round tenons can be enlarged to fit into oversized mortise holes by either wrapping the tenon in a glue soaked plane shaving or by expanding the tenon diameter with a wedge. Mortises that are cracked or split can be re-glued as long as the wood closes snugly so that the glue will stick. If not, a new piece of wood should be spliced in and the mortise re-sized to fit the tenon. Dovetails Dovetails are another classic joint that form a mechanical lock in addition to the glue bond from the mating wood surfaces.