My friend changed her tyre with the bike upside-down, struggling to slot in the axle and the derailleur just fell off from pressure with her bare hands.
She is far away so I need to understand what tools to take before I go there. There's a weird bolt with a blue end and a flat washer, probably the derailleur was holding on mostly from the force of being pinched by the axle bolts. It seems to have been monkeyed with weird bike theories. What's going on and how can I mend it?
That's a generic 1980's style claw adapter for a derailleur - quite common for its age.
The silver circle with a segment cut off - that has a round boss on the other side which sits inside the dropout slot. The side visible in your photo faces toward the cassette, and the bolt with the blue dot clamps the hanger to the dropout slot in the frame.
The second thing holding the hanger to the frame is the bike's axle so the open mouth on the hanger lines up with the slot.
Should look something like this:
Tools - you should get away with common hand tools, so screwdrivers, two adjustable spanners, maybe pliers etc.
In theory, the derailleur is clamped into the dropout by the silver stepped nut, with the flat towards the axle.
In practice they don't really work, and the axle nut is what holds everything together - until you take the wheel off. The blue is probably threadlocker, to try and get round the looseness. These and other similar designs (a tapped hole at the dropout instead of the nut) often strip the thread on that little nut or its screw, because the holding force you want is more than you get.
I never bothered actually solving it, and on the bike I had that did this, just relied on putting the derailleur back onto the axle whenever I had the wheel off. I ran a tyre liner and a fairly thick tyre and tube, so didn't get many punctures on my short urban rides.
My commuting toolkit would deal with this: a pair of spanners for the axle nuts, a smaller spanner (probably 10mm) for the hex head on that little bolt (a screwdriver fits but doesn't give enough torque), and disposable gloves as you'll get filthy. I suggest slotting the wheel in with the axle nut as loose as it will go without falling off.
