A gamma ray measurement is always fun, but it is slightly hazardess. A better one, is the electron spin resonance measurement. with a good regulated power supply you should be able to measure the Bohr magneton almost exactly. The biggest problem of the experiment is the alignment with the earths magnetic field.
Well, those experiments which came under the 'nuclear' section of my first year lab all tended to go horribly, horribly wrong, despite them being relatively simple:
Measurement of the Charge/Mass ratio on an electron - (Probably the most interesting one I did, and went pleasantly well)
A recreation of Millikan's Oil Drop experiment - (interesting for the write up, but desperately dull to conduct. Also the error analysis was just infuriating. "what was the error on this?" "Er...about two foot, or something")
Electron diffraction through a crystal lattice - (again, great for the write up afterwards, the experiment consisted of sitting in front of a phosphorous screen and talking about what kind of sandwich you'd offer Jesus if he came round your flat. Answer: Cucumber, surprisingly).
Thermionic Emmision of Tungsten - (Net result - an insane prejudice against wheatstone bridge circuits).
There were others, but I can't remember them and thus conclude that they were far too uninteresting to ever be unleashed upon innocent young undergrads.
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