I was running my mouth again and now someone wants a citation. I claimed that you can force a non-measurable set of reals into [some] model of ZF without introducing choice. I think I can write the proof, but that's not a citation.
Namely, you can do forcing on a model of ZF if you have choice in the ambient universe, right? So you just take a model of ZF¬C and add a random real, which forces there to be a non-measurable set.
Maybe it would be easier to start with a model of ZFC with such a random real, and kill Choice by building a permutation model whose set of atoms is uncountable and less that the size of the continuum?
At any rate, I'm excited to see what is already out there.