My dad has the pnuemonia shot himself along with the flu shot. He didn't experience the same thing your mother did but he did contract a bad dose of flu which put him in hospital with a severe lung infection. Im really not sure what it could be. But since its hot to the touch that would make me think something is going on. Possibly an infection of somekind or the nurse did miss the vein like your mother thinks and caused bruising underneath the skin. It could be any number of things or its possible you mother is having an allergic reaction the the shot. I have heard of reactions happening but I dont know of the symptoms.
However, if the swollen area is hot to the touch, discoloured, bruised or itchy I would take your mother to the doctors asap and have this checked out incase its a blood clot. I dont want to scare you and I hope I haven't. But its best to rule it out. I wish you luck!