I'm almost certain it's a scheduled system task that's causing the problem. Not necessarily an update - that's just one of several dozen possible events that could fall under the general category of system tasks.
Go to the Start menu and type "task scheduler" in the search bar. Open the task scheduler and take a look at the list of Active Tasks. See what's scheduled for 1:00 AM in the "triggers" column, and you'll probably narrow it down to one or two possibilities. I do think a hard drive defrag is one of the things that is typically scheduled at 1:00 AM by default, so that definitely sounds familiar.