A variable should be a target for elimination under certain conditions. Check both the variables loading(s) on a factor as well as its communality. If the variable loads more than one factor, it may be a target for elimination. However, there are cases when cross-loadings are expected. Now to your question. If the communality is low this suggests that the variable has little in common with the other variables and is likely a target for elimination. Look to the WISC-V as an example. The Cancellation subtest has a low communality, a low general factor loading and struggles to align with a group factor. If one were a psychometric purist then a case can be made that this variable/subtest be eliminated.