Since you say that the wood under the hinge and around the hinge screw holes has split and gone soft it pays to check the wood of the door jamb adjacent to the hinge mount location as well. If the wood there is not sound then you are likely looking forward to some more serious repair such as replacement of the jamb side board or the whole jamb.
If the adjacent wood is still in good condition it may be possible to cut out the the piece under the hinge plate and replace it with a new block of wood. Edge gluing of the block of wood into place will require a very close tolerance fit and ability to clamp it into place to get the strongest joint possible. The best side for the wood to bond well with the new piece is along the grain so make sure the insert piece has the same grain orientation as the jamb. Then you can clamp across the jamb and to do so it would be necessary to remove the casing trim along the hinge side.
I would not drill out and chisel the the opening to shape as it would be a lot of work to get the tolerance necessary for a decent wood glue bond to the insert piece. Instead I would use an oscillating multi-function power tool similar to that pictured below. Using the plunge cutting blade like the middle one pictured it would be easy to cut out the damaged piece from the jamb with nice straight and square cuts.
Once you have the new piece glued in place and you go to remount the hinge make sure to drill appropriate pilot holes for the hinge screws so as to avoid splitting the insert wood piece.