You could try just using deep standby (uses virtually no power) and see if that makes a difference. The box isn't really supposed to be completely switched off daily just for a prolonged period of time. Which backup version are you on (zip file date)? I'm running the latest (2015-06-23) and haven't encountered it.
Things you could check /do:
Is it possibly router settings - do you have dhcp enabled on your router(it should be!)?
Have you reset your router lately? (perhaps it isn't re-using IP addresses and has run out of allocatable addresses if not re-using the IP's and grabbing a fresh one each time you power your box)
Try a network test from MENU, setup, system, network, device setup?
Do you have two zgemmas by any chance and both running my backup? (They will share the same mac address - could this potentially make the software disable the port I don't know??)
Re-run the network wizard... you never know!
Sorry I'm really just clutching at straws as I've never come across this but all may be worth a go or get you closer to a solution. Please let me know if you get to the bottom of this....