What to do when an omitted level-2 predictor is associated with a level-1 predictor? Raudenbusch and Byrk (2002) write on p. 261:

If an omitted level-2 predictor is associated with a level-I
predictor, the coefficient for that level-1 predictor will be
estimated with bias. In this case, we have $Cov(u_{qj}, X_{q'ij}) \neq 0$. This problem can be solved, however, without elaborating the
level-2 model. Any covariance between a level-1 predictor, $X_{qij}$
and a level-2 random effect must operate through the covariance
between the group mean $\bar{X}_{qj}$ and that random effect. Such a
covariance can be eliminated by group-mean centering the level-1
predictor.
Alternatively, inclusion of $\bar{X}_{qj}$ as a covariate in each
level-2 equation will eliminate any confounding between $X_{qij}$ and
omitted variables at level 2.
These two approaches suggest a specification test for the effect of
omitted level-2 predictors on a fixed level-1 coefficient. One runs
the model with and without group-mean centering of the level-1
coefficient. If the fixed level-1 coefficient associated with that
level-1 predictor remains essentially unchanged, the model is not
vulnerable to this type of bias.

They don't seem to pass any judgment on which of these two approaches should be preferred if the model is vulnerable to this type of bias. What should drive such a decision? Is one of the methods generally better?
Raudenbush, S. W., & Bryk, A. S. (2002). Hierarchical linear models: Applications and data analysis methods (Vol. 1). sage.
 A: A little late to the party, but I thought I'd provide some perspective. In this section of the text they are talking about something economists call endogeneity or subject-level unmeasured confounding. These are fancy words/phrases for the situation where a correlation exists between predictors and the error term (of which there are at least two in a multilevel model).
The possibility of endogeneity is the reason that economists tend to prefer so-called fixed effects models, in which they treat the clusters they have in their data as the universe - they are only interested in making inferences about that group of clusters. Practically, fixed effects models include a set of 0/1 indicators for all clusters as predictors in an OLS model (although R, Stata, SAS, etc. have special packages or programs for these models). The challenge with these models is that group-level variables cannot be included so if one's theory is about a group-level variable, they will have trouble testing it in the fixed effects framework.
In terms of the two options proffered by Raudenbush & Bryk, I don't think either one is more highly valued than the other. Each strategy "protects" the level 1 variables from level 2 endogeneity, but any level 2 predictors are still subject to endogeneity concerns. It depends a bit on your modeling goals. It is suggested by Enders and Tofighi (among others; see this nice summation by Jason Newsom) that if you are going to specify random slopes for lower-level variables, then group-mean centering is helpful. So in that case, you might group-mean center all level 1 variables. If, on the other hand, you are not modeling random slopes, then perhaps just including the level 2 means of of all level 1 variables in the model is better. At least there, you can look at the tests of significance for the group mean coefficients to see which variables might be problematic.
