I have been building for a long time, so I know where floor joists can be bored or notched without compromising their strength. Unfortunately, I have a client whose floor joists were hacked at by someone who didn't know the rules. Her floor joists were notched by someone installing electrical cabling IN THE MIDDLE OF THE SPAN ON THE BOTTOM OF THE JOIST!!!!!! I know that I can reinforce the existing joists by sistering new joists to them, but I'm curious - is there any reason a carefully installed and glued patch wouldn't do the same trick? It's been my experience that wood glue is stronger than the wood grain around the joint - at least as far as shear values are concerned. But the end-grain to end-grain connection is where I worry that the glue would not hold up under tension. Does anyone have an insight as to the efficacy of a patch?
Gluing a patch in place like as shown below will not add much to restoring the original strength of the joist.
If you were to attach a longer piece like shown in the second picture and glue this in place for the full length of the piece a whole lot more strength will be added to the joist.
Note that if the joist has started to sag down due to the inappropriately placed holes or notches you may want to try jacking the joist back into its original position before applying the glued on pieces.
The length of the piece that you would use is going to be dependent on an number of factors including the overall span if the joist and its top to bottom height. The material could be a variety of different things including 2x4, 2x2 or even strip of 3/4 inch plywood. You could also consider gluing on a strip on both sides of the joist. When gluing make sure to clamp the pieces to the side of the joist with good pressure to make sure the wood joints are tight.
For an end grain glue joint to properly hold, the joint would have to be some sort of long scarf joint, just sawing it off and end gluing isn't enough, end-grain pulls out and yes, the glue is stronger than the end grain. You're not going to gain much with the extra work and it's probably not approved for use in this location.
Along with what @shirlockhomes says in his comment, slap some wood glue on your repair piece and sister it to the joist. The wood glue will be a stronger joint than the nails.