We used to get the same periodic reports of webmail logouts, but haven't gotten one in 2 years. It was always the time settings being off between the PC and the server, even just a few seconds can trigger the issue. You can reproduce the condition by manually setting the wrong time on your PC and then log in to webmail.

You can also reach out to support and they'll send you instructions for capturing the logout activity with Fiddler and with the browser's developer tools.

Check and see if the end users network hardware controller is set to sleep on low power. If the user is not actually using the machine or network and it is just sitting there, the OS can literally turn of the network adapter to conserve power. Personally I think that is a stupid feature, but there it is.

Curious on this one too. We weren't having any issues with logouts until we got upgraded from 8930, now it seems like we get randomly logged out from time to time (I've checked the above settings). 

Before the update, i myself not sure about others, would stay logged in for weeks on end on my laptop, regardless of powering down. Shouldn't the "Remember Me" at login keep the user logged in via cookies or something regardless of network or power situations? i know it's a security issue, but some users only access via offices or secured devices, so clicking Remember Me and then still having to log in tomorrow is a bit annoying. The kicker is, it's not consistent. Sometimes, again myself, i can go 2 or 3 days before i get prompted to log in again, and sometimes i get prompted twice in a single day.

This recent resurgence of this issue is one we have seen from a few clients, but what makes it difficult to track down is that it is not all clients either. And of the ones that do experience it, like Richard is saying, is inconsistent with currently no known reliable replication steps. We have had a couple of our testing environment do this but nothing stuck out on the client or server side logging and has only happened a fewer times than you are reporting. Any additionally things that can help replicate this issue would greatly expedite our troubleshooting efforts.