Enroute Controller described a below authorized altitude event when failing to review route filed vs. altitude; listing both weather distractions and low Terrain Alert Volume (TAV) settings as contributing to the event.

Narrative

I was working an IFR aircraft descending to 11;000. I was also working a VFR aircraft. The VFR was more than 50 miles south of his intended course trying to avoid weather and having difficulty seeing terrain at night. I advised the VFR of weather reported at airports along his route and suggested a safer route. The IFR aircraft inbound advised the VFR to remain south of his destination and proceed west awhile before turning north. At this time Tower requested IFR departure for a BE20. I released the BE20 to 12;000 as requested and stopped the inbound at 13;000. The Terrain Alert Volume (TAV) in the Center had recently been adjusted and I thought the route of flight would work for the new TAV [setting] but I would check it when he was airborne. The IFR inbound cancelled his IFR rather than be delayed into the airport. At this time I should have climbed the BE20 to 13;000 which is always safe going to his destination but neglected to do so. The BE20 entered a 12;900 TAV at 12;000. I advised the BE20 to maintain 12;000 while in controlled airspace. He answered; 'Maintain 13;000 while in controlled airspace.' I decided it was easier to allow him to climb than to correct him. I should have cleared the BE20 short to the VOR than checked the route to make sure it cleared all terrain.