Sounds like the plastic shift cup is shot (hopefully it isn't the tranny synchros). Rebuilding a shifter is pretty cheap compared to a spare gearbox. Check RealOEM.com for your car... I don't know if you've been there, but it's a great resource. You can find BMW part numbers (even for really rare Euro parts) & even do cross references to see if other BMWs carry your part.
There are two manual tranny linkages, one is sheet metal and the other is tubular. IIRC the sheet metal one is older, more likely for your e30 M10. I'm not the expert on tranny swaps, but it's my understanding that all BMW transmissions can use either linkage but parts don't swap between linkages. So if you like the one at the yard and it isn't the same type, grab the whole thing.
Looks like your son's is the earlier type,
click here for the diagram. I'll bet the cup (part #4) is the culprit. Might be
#7 here on the shifter plate too.
If they're that cheap I'd definitely just order them instead of heading to the yard. Those prices aren't totally accurate, check BMA or Pelican for a more up-to-date price.
Many transmissions will fit your son's car too, but they'll need the right drive shaft to fit. I'm pretty sure it's a Getrag 240, so any M10 or M42 e30 would have the parts you need. 260s are longer and I'm not 100% sure they fit correctly.