You could try running sgmon.exe which will be found in the Backup Exec directory to collect some debug logs.
Under Capture choose "Select All" then hit the green Start Capturing button.Make sure to stop the debug collection soon after the BE Gui starts responding again. Then have a quick review of the logs produced on the screen - although there is no easy answer when reviewing debug logs, its worth seeing if anything jumps out at you (eg exceptions/errors/failed messages etc)
Also try and make a note of the time the console came back to life, and cross-check this time with the debug logs to see if certain messages you were getting when it was "asleep" stopped once it woke up.
If you get a chance to repeat this procedure a few times it may give you a better chance of spotting something relevant.
Apologies that I have no better suggestion than this at present..!
