Not much context here so I could be wrong, but I read "in an attempt to hold on to what I could of him" as "in an attempt to keep him (the professor) in as good a health as possible". It sounds like the old professor's health was failing badly, and the speaker intended to give the professor a massage to try and help him as much as possible. ie. to figuratively 'hold on' to the professor's health and stop it 'leaving'.

“In an attempt to hold on to what I could of him” = “in an attempt to retain/maintain (some connection with him) (some relationship with him) (some memory of him).” “Of” means “related to” or “in connection with” him. Example: “In attempt to hold on to what he could of his (dead) mother, he kept photos of her on his wall.”