If the joists are not strong enough as they are then I would forget it.
However in most cases placing boarding accross the joists will provide a sound floor.
The cheapest oprtion is probably chip board and I believe that this can be bought in relative small sections just for this purpose.
This then need to be cut to shape and secured to the joists. I would suggest screwing the boards down as nailing could lead to cracks in the ceiling below. Pilot holes and an electric screw driver a must!
Joists can normally easily be sistered. This is when another usually larger one is nailed or screwed to its side. This stiffens a floor quite a bit. Depending on your existing joist, load and span would depend on what side you would go with. The draw back of this is your floor is raised above the original location making your ceiling height lower and may mess up your flow on the up stairs a bit (all should be considered). What ever you do, use the screws. Nails unless using a nail gun will for sure stress your lower ceiling causing damge.