For the CFM56 engine model CFM56-5B3/P, what does "5B3/P" mean?
It's a more specific designation of the general engine. There are lots of variants of the CFM56 in your case the 5B3/P variant:
CFM56-5B3/P: Same as CFM56-5B1, except for increased thrust rating and redesigned high pressure compressor, high pressure turbine, and low pressure turbine.
The type certificate has the specific info for it as well. There is also lost of info on wikipedia under the -5B series section.