If it is a C-41 black and white film like Ilford XP-2 or Kodak BW-400CN, or whatever the new Fuji one is called, it should cost the same as color developing and prints.
If it is a single-layer ("standard") b/w film, my local lab (A and I in Los Angeles) would charge $8 process only (b/w, C-41, or E-6), $15 process and proof, and $20 for process and single prints ($7 per roll extra for a full set of duplicate prints). I don't like the way their prints from b/w film look, as they are on color paper and are not full frame. (I do like that they process in a Refrema Dip-N-Dunk, however.) Therefore, the most I will do is process and proof, and that is a very rare occasion in which I will lab process b/w film; usually when I have a large batch that I want to be quickly processed in the same manner.
Unless your son needs to see a print of every single shot he took, I would not bother paying for prints. At the most, I would pay for processing and proofsheets, and usually just processing only. The prints won't be optimized for contrast and density, they will have a color cast, and they won't show what your son composed in the camera.
Some true b/w prints are available, through Freestyle or Ilford. I haven't tried it yet, however, and it is still quite expensive.
IMO, if you are not going to do the lab work yourself for b/w, just shoot Kodak BW-400CN. It is made to be printed on color paper, unlike the Ilford XP-2, which is a similar product, but made to be printed on b/w paper. (If your lab can print on actual b/w paper, by all means shoot the XP-2 instead.) Any local cheap lab can handle the C-41 b/w film, and this will likely give you the same or better results than giving a standard b/w film to the same lab.