Disclosure: I am a current Motorola Home employee, but I have no insider insights into the split planning process.
This is speculation on my part, but it seems like both halves of the company may take on new names. However, the Mobile Devices/Home company will own the Motorola brand and license it royalty-free to the Government/Enterprise/Networks half of the company. (This was included in the public announcement on Thursday.) It will be messy, but it will allow both halves to carry on for a while while (potentially) building new brands.
It will be an interesting ride. The Home and Mobile Devices units have been part of the same company for a while, but never seemed to collaborate much. I'm guessing that will be changing, along with some of the culture and organization in Home. I expect it to be somewhat painful, but worth it.