There seems to be a handful of papers on single-subject resting-state fMRI data but not for EEG data. Is this simply because it is too variable to be done? If not, what are the things to watch out for in data collection and in analysis?
beginning with a shameless plug: we recently carried out a study looking at EEG oscillatory networks during both rest and task. we found that both types of networks, in several frequency bands, were sufficiently stable to recognize individuals across a 6 month period. so I'd say it's pretty reliable. (preprint available in attached link.)
other than that, it's hard to give specific recommendations without knowing what you're hoping to find (beside the standard don't move & relax instructions)...