How to repair a floor joist which has a hole drilled less than 2 inches from the bottom

You should shim under the loose joists with steel shims, not cement, because cement causes wood to deteriorate. Large galvanized washers and small steel plates can be used as shims. The framing block in between the joists are called bridging and in an old house I would space them about 4' on center. You can off set each row by 1 1/2" which will enable you to blind nail through the joists which minimizes toe nailing. Use 3/4 tongue. And groove ply glued and screwed to the joists and bridging. I would use pl400 for glue and 2" exterior screws spaced about 8" apart. If a second layer of ply is desired you should also glue and screw it the same as the first sheet. Try to hit the joists. Use a chalk line to mark the joists. If a solid wood finished floor is being installed 1 layer of 3/4 plywood is all you need.if a ceramic tile,stone or wood laminate floor is going in use 1/2 ac plywood over the sub-floor.

I spoke to my structural engineer and the answer is that it depends on the depth of your joists. In my case, where I have 2x10 joists there is no problem with a small (1-inch diamater) hole on or near the area where the joist is bearing on its support beam. Basically the same notching and boring rules apply at the end of the joist: 2 inches from top or bottom, etc..