I've looked around the files this app uses and can't see a way to manually set a new name.
If you think your family would find it easier to remember that a larger number meant the living room DroidBOX room, you could:
Change the WiFi/Ethernet configuration on one or both DroidBOXes to use Static (rather than DHCP) IP addresses. (Let us know the models you have if you need a hand with this).
Once the DroidBOXes have something like 192.168.1.100 and 192.168.1.230 (the 192.168.1 part will depend on your home network's setup), in the DroidBOX Share (client) app on your phone or tablet, if you tap the screen in the middle at the top (where the currently connected device is mentioned), you'll see both DroidBOXes listed with the DroidBox-7248 type label, but also the DroidBOX's IP addresses. So if they are happy remembering the DroidBOX with a bigger number is the living room, they'd know which they're about to connect to.
That will make a lot more sense if you try it out, compared to reading that text!