My floor joists are 2x10s, 16 inch on center. Before I drywall the basement ceiling, I would like to stiffen the floor joists to eliminate bounce, by installing new pieces of 2×10 perpendicular to the existing joists. How far apart should I put the new pieces of 2×10? I am trying to find the balance between having it work, and overkill.
It sounds like you are asking how to install bridging or blocking, which can be an effective way to stiffen a floor structure. The answer depends on the span. For example, if your home is 24 feet wide, and has a center girder, then you'd likely have floor joists spanning about 12 feet. At that length, I'd be surprise if you do not already have bridging or blocking installed. However, if you don't, you could split the length of the span in half and place bridging that at the 6 foot mark.
Bridging is a sort of criss-cross design made from either wood or pre-fabricated metal. It is easiest to install this type of bridging before the subfloor is attached. Or it can be made of solid blocks, constructed of solid 2×10 (as you suggested). A tip is to offset the blocks so you can nail through the floor joist and into the end of the blocks from both sides. See the illustrations in this article about floor joist bridging. Finally, you also need to be very careful working around ducts, pipes, wires, or other important house elements.