mmm... You can try a short-cut to System Flush in hope, and/or Fixdisk, but if that doesn't work you'll just have to worth through the diagnosis systematically. I see no particular reason a firmware reinstall is any more likely a solution as any other.
The network connection is wireless. I've had dnla-filter installed for ages and it's only in the last week that the random reboots have been happening. As far as I know nothing has changed on the network but I will triple check with the kids just in case one of them is 'trying' something. Reading through the link that Black Hole supplied, I was going to give the reset DLNA database option a try first.