I have encountered this situation on several occasions in the past, ie. friends, relatives,fixeruppers and the very best solution is: Sister 2" x 10" along side the existing 2 x 8's and glue with construction adhesive and nailing with 16p nails every 10". you only have to do this on the central part of the floor. That is you do not have to go to the very end. The headroom you will lose will not be any greater than installing 2 x 4s and will guarentee that the bounce will be eliminated.