Software wise it'd be both individually programmable and have access to some sort of thingiverse equivalent with actions that others have dreamed up. Otherwise it would not really need an internet connection at all, except perhaps to the local wifi to talk to other more-or-less automated systems in the same house.
For applications, some of the ideas I had while typing the rest of this comment:
- If there is a package outside the door, open the door, pull the package inside, close the door again.
- If the mail has been delivered (for houses that have a front door with a mail slot in them), use the vacuum attachment to pick up the mail and deliver it to a central location (desk? dinner table? kitchen?)
- If the dishwasher is done, use universal gripper to pull it open, open the closet and put the dishes away. (Or into a drying rack, whatever).
- For most people I know the washing machine for dirty laundry is someplace away from the bedroom. I don't think it'd be reasonable with current tech to expect robot folding of clothing but at least it could pick up dirty laundry from various places where it's collected and bring it to the washing machine so people only have to come in and turn the machine on.
- Depending on how good the gripper is and how well adapted my coffee machine is to it, perhaps it could pick up a coffee capsule and prepare it while I'm in the shower.
- Once per day it would make a round around all the bathrooms and refill the spare toilet paper roll holder if there is only one roll left.
- Maybe it could wipe down sinks and stuff regularly too.
All in all, it'd be useful several times per day I think. Not doing anything I couldn't do myself but definitely taking care of various simple chores throughout the day. It would need pretty good sensor coverage as well of course. Finally, I realize that with current tech this would be prohibitively expensive.