Step 1: Prevent air transmission between the rooms. Sweeps for the door or a heavier door; look at the heating registers and windows; things like that. You're saying the sound is muffled, which implies the high frequencies aren't making it in, so I don't think this is the major problem.
Step 2: Damp the transmitted vibration. Put big heavy bookcases on the wall between the two rooms (which side doesn't matter so much). Blow heavy insulation into the wall between. Hang a heavy quilt on the wall.
Step 3: Break up the sound in the source room. Use natural diffusers (heavy sofas with heavy foam pillows work great, or the aforementioned bookcase with loads of stuff on it), or buy foam diffusers to stick in the corners/edges of the room.
Step 4: Give up and wear sound-cancellation headphones...or install a white-noise generator in your office (either a fan-type one, or a water feature if you like more 'natural' sounds).
See also How Do You Reduce Floor Sound Transmission for more details.
I wouldn't tear the bathroom wall apart... I'd tear the bedroom wall apart. What you are experiencing is the noise traveling through mechanical connections to the wall (screwed to studs) and the noise from the water passing through thin copper pipe (probably the Type M pipe rather than the thicker Type L).
The only effective way I know of is to insulate that wall with Roxul Safe and Sound. It's an insulation designed for noise.... and noise only. It doesn't burn. To go the added mile, use resilient channels before hanging the drywall. The demo is probably a DIY, the installation of the Roxul is DIY, and the drywall.... call someone. For a few hundred bucks the repair will disappear.
Edit:
Blown in insulation will not work... blown in is for warmth, not sound. It's the sound transfer you're trying to conquer. We once built a house and insulated the walls with thermal insulation, thinking just like you are. It didn't work. We had the type L pipe, all the plumbing was fastened with cushioned fasteners. We thought we had it knocked. Turned out, it was a waste of money.
If you truly want to fix it, just take down the drywall and do it the right way with the Roxul. Going into a wet wall from the bathroom side is a suicide mission. Building an offset wall will create a door opening that's 10"... with a door that has a 5" jamb. Then there's the carpet... or other flooring surface. It sounds like a difficult job, but it's probably only one weekend.
Otherwise, just buy your son some foam ear plugs at Costco and call it a day.
Best Answer
You can get acoustic plasterboard (UK site). It won't block out all the sound (as pointed out on the site) but will reduce it - hopefully to a more tolerable level.
Stick this to the wall and then skim/tile on top.
You will lose a bit of space from your room, but it's only one wall (the party wall).