Mixed-effects modeling with crossed random effects for subjects and items
R.H. Baayena, , , D.J. Davidsonb, , D.M. Bates,
doi:10.1016/j.jml.2007.12.005
Has a very clear account of testing especially the difficulties associated with obtaining p values for random effects - the 3rd author is the originator of the lmer function in the R.package
You can use the function `anova` to compare models with different random effects structures using chi-square difference tests (see attachment). By using stepwise model comparison, you can attach p-values to different changes in random effects structures.
Along with the other resources about p-values and lme4 in the other responses, see also Bates and colleagues work on trimming down maximal random effects structures: http://arxiv.org/abs/1506.04967