Does timing of PhD application submission matter? Is there any benefit of applying in October as opposed to late December right up against the deadline? Do committees just meet after the deadline once all reviews are in? This is what I thought, but I have heard from a few professors that they sometimes look at applications as they roll in, and so it may be advantageous to apply early because they could remember yours. 

Edit: One thing that I would have to weigh applying early against is the fact that I work at a research org, so waiting an extra 8 weeks or so could allow a new publication to come out and add that to my CV, or I could get on a new project in that period which would be relevant and look good to the programs I'm applying to.
I doubt that it matters in a formal sense at very many institutions unless the school used "rolling admissions". However, it is possible that if you apply early and there is some flaw or lack in your application that someone, even a clerk, will catch it and give you a chance to correct it. 

Sometimes a department head will try to shepherd early applications since it is his/her responsibility to make sure that there are enough applications to build a new cohort. That is separate from formal acceptance, however. 

It is also possible that an especially "juicy" application will be passed to a professor who might be looking for a certain kind of background. But again, that is separate from acceptance. 



In rolling admissions, students are accepted or not as the applications come in. The deadline may mean only that the process stops at that date.