After fiddling around with this more, I believe I have found the problem. I have found that there is a registry key WebClientComputerName located at HKLM\SOFTWARE\ESET\ESET Security\CurrentVersion\Info that is not updated to the new computer name but contains the old computer name. The key must be protected by ESET as I am unable to change it manually even having permissions. I found that if I uninstall ESET Endpoint Antivirus and re-install, the key now has the correct value and EBA now shows the correct Device name.
One other thing I noticed was there was a computer we renamed two days ago that the value did update in EBA. In looking through all the licenses in EBA that I know we renamed the computers in the last few weeks, the computers that had ESET Endpoint Antivirus 8.0.2028 appeared to update the Device name in EBA correctly. Computers that were renamed with ESET Endpoint Antivirus 8.1.2031 did not. Uninstalling 8.1.2031 and then re-installing fixed the bad Device name in EBA.