First question: what's your area of physics? Thermodynamics, hydrodynamics, aerodynamics, nuclear physics, material science?
Sufficiently low-level that I could probably swing most of those things if I tried hard enough. Nuclear physics probably won't fly due to budget constraints, although I could try, but I could probably do something in pretty much ask the other fields.
It does rule out much of the super-cool stuff because it's relatively low-level (not helped by the hands -on requirement either) but there's also a lot that isn't.
"low-level" doesn't mix well with "reasonably serious experimental project"...
To clarify, I mainly mean that I'm obviously not going to be doing bleeding edge research 'cause that's all super pricy and I don't actually have a doctorate.
But it's able to be significantly beyond the level of measuring g or whatever. The spec is literally 'whatever interests you'.
As for fields that interest me, sure, there are lots. There are a handful of projects I already kind of have in mind. The point of this is in case there's something I haven't thought of that's actually pretty cool, since obviously I won't know about it.
Build an electromagnectic boat! Or just a thruster. I'd bet you can make a (model size) one that works well enough to demonstrate the concept. It's really just a railgun with water, you could probably make a measurable current with stuff from the hardware store, but a few relatively expensive magnets the university can provide.
You could probably also build a nuclear reactor, like that one kid, within the university's budget, if you can get appropriate shielding/safety approval.
Huh, the electromagnetic stuff is neat.
Budget isn't necessarily going to be the issue with building a nuclear reactor.
Their budget is probably more generous than their paranoia.