"The Rig" is the nickname given to a haptics device that was built by Philip Linden and Andrew Linden with the original intent that Linden Lab would be a hardware company in the haptics field. Second Life was then created as the software to go along with the hardware, but eventually took off on its own, resulting in Linden Lab leaving the hardware field all together.
The rig still exists to this day, sitting in pieces in a box in the offices, since hardware is not currently a major pursuit for LL there is little development being done for the rig; however it is tinkered with occasionally. The rig is not something that is worn (such as a Head Mounted Display), but rather a construct that is set up around the user in multiple pieces, each piece being a different way of giving input. The user then manipulates the hardware to interact with the action on the screen in the same way most haptics devices work. The nickname most likely comes from its "rigged" appearance.
Linden Lab has stated that at some time in the future they may return to hardware but there is no current plan to do so due to costs involved and they enjoy working on Second Life. There are also many other haptics companies that dedicate full time to the development of hardware and LL could possibly just use one of those as a peripheral device for SL.