Where to start? Here's a few things.
Service history is king here, if the car has been with the present owner for a good many years and there's lots then a lot of the below wont be an issue. If its a short term owner with little history then buyer beware. Hope this helps.
Ends of the sills Front and Rear, they corrode badly. (relatively easy to change as they are bolt on cosmetic parts but not so easy to get hold of new ones)
Rocking drivers seat. The plastic pads in the adjuster screws wear / disintegrate. Easy(ish) cheap repair.
Damage to drivers seat bolster. Its a wear and tear issue.
Shift Pins. Check the gear stick lines up with 3rd/4th when in neutral if not then the pin shims are worn and its a gear box out repair however it wont stop the car being used, its just annoying.
Check AC, Heated seats, Electric roof electric mirrors and windows all work (if fitted)
AC if its not working it may well be more than just a re-gas, the condensers are vulnerable and can get holed. Repair approx. £600ish
Check the carpets are not wet push down hard several times to see if any dampness comes up. Its an issue!
Check Boot for water. If you can check the battery box area and to the left of the left rear light remove the side carpet (easy to do) stick your hand into the area alongside the light, right to the bottom, if its full of water then most likely the aerial grommet is leaking. Just get some cash off, its an easy cheap fix.
Check Boot welds. Remove the boot carpet completely, its designed to be lifted out. You will see a row of spot welds running across the boot. If any are pulled
I would walk away this shows the diff hanger is breaking away underneath.
Engine for oil leaks. Not usually an issue is its been well maintained.
Rear Tyres. Check inside edge of both tyres, if severely worn compared to the rest of the tyre the car has either been over lowered or the suspension is very worn (Z3s do tend to wear here to a degree)
Check Headlight switch. The switch is also the rheostat for the dash lights (pull the switch to turn the lights on and twist back and fore checking the dash light dim/ brighten gradually) these fail and new ones are £100+ish
Heater. get the car to normal driving temp and make sure the heat works in all positions, it should blow VERY hot, the Z3 heaters are fantastic if working correctly.
Excessive diff whine. The diffs are not the quietist but it shouldn't be very loud.
Suspension. If its original or very old it will most likely want changing. Lower front wishbones and rear donut bushes would most likely need changing DIY if you are handy with the spanners. Rear beam bushes will be knackered if original, difficult to check. It wont stop you enjoying the car but handling will be compromised. Rear beam removal to repair, not easy DIY job. Shocks are easy DIY job and not excessively expensive.
Brakes. After a test drive check each wheel If excessively hot then a brake is sticking. Common with badly maintained Z3s
Dash warning lights. Check all the lights come on with ignition then all go off after starting.
Rear window condition and surround. If its in bad shape then a new window / roof is the only option. New fitted roof at Jack Smiths in Swansea approx £400
Temp gauge. The car should run with the gauge smack bang in the centre, if not then its possible it may only be a thermostat change needed.
Tamper Dot. Check by the oedometer, if there's a dot then the cluster has been changed / messed with. There needs to be an explanation if this is showing.
Central locking. Check it works on both doors and boot.
Good luck, hope its a good car