What is the best way to patch 3" holes in plaster
This question is very similar to this question:

How Do I Fix a Hole In Drywall

There are some good answers posted, so it is worth a read.

To build on those answers for this particular case, I would recommend the medium size hole fix.  The only difference is since the hole is on the small side, trying to screw the new 3" circular piece may crack or break it in half or in pieces, so my recommendation is to glue it to the strapping or backing material using gorilla glue or similar.  Just make sure the new piece doesn't stick out of the existing wall.  If you can make it slightly less than the existing wall, that would be optimal, and then you can just fill in the depression with mud/spackle.

If the plaster is painted, you can use mud to fill in, it doesn't have to be plaster as the paint will cover it over.  If it is unfinished plaster, then fill in with plaster as the finish will be the same.  It looks to be painted, so standard mud/spackle should work out fine.