bootstrapping in regression

Others have confirmed that you use the predicted values plus permuted residuals is the new y variable and also referred you to some other articles.

On the question of does this work for mixed effects models:  That is a good question, and it depends on what question you are trying to answer and what assumptions you are trying to make.  The mixed effects model is more complicated in that you not only have residuals that you are permuting, but possibly also random effects, depending on your question(s) of interest.  Then to further complicate things, you need to take into account any correlations between the different residuals/effects.

If you can work out a reduced model of interest under your null hypothesis, and see how to permute the other pieces in a way that preserves the correlation or works with assumed orthogonality.  Then it should work for you (but not be simple).

I would suggest that you try doing a bunch of simulations where you first create data sets that follow your null hypothesis (reduced model), then do the permutation test on them.  If everything is working correctly, then the p-values should follow a roughly uniform distribution (if not, then the permutation test is not working for your situation, your assumptions are not holding, or something else is messed up).  Doing the simulations will force you to think about all the pieces that go into the analysis and how reasonable your assumptions are.  If this works, then try simulating under the alternative (full model) to see what type of power you have to see the difference and compare that to other approaches.

Hope this helps,