HP 6310 printer shows up as Unspecified

!!!!Bluetooth mouse!!!!

Could you tell me what that laptop has listed in the Device Manager for Network Adapters?
I'm guessing there are 2 - One Atheros 802.11b/g/n Wireless Network Connection, and one Bluetooth adapter of some sort.

Yes, I'm thinking that the Bluetooth adapter is causing it, triggered by the mouse somehow. But I am only speculating now - I don't play with Bluetooth devices.

But I think a good experiment would be to disable the unused devices in Device Manager and then try installing the printer (after cleaning up the OS from previous installs, of course).

The Marvell Yukon 88E8057 PCI-E Gigabit Ethernet Controller is your wired LAN controller - keep it.
The Atheros AR9285 Wireless Network Adapter is your wireless adapter - and the one the printer would use.

All the other adapters could be disabled to test. I'm guessing that the printer is "confused" over which wireless adapter to use.

You will, of course, need a standard USB mouse to do this, or just use the touchpad.

If the printer does install you could enable the devices again and test, or if you don't have any use for them - leave them disabled.
If you Remove these devices they will most likely just reinstall on the next startup, so unless you can disable them from the BIOS settings it would be best to just disable them. You may have a use one day.

EDIT: More info-
Microsoft Virtual WiFi Miniport Adapter
Your call.