Center Controller reported allowing an aircraft to fly below the minimum vectoring altitude after inadvertently becoming distracted with other tasks.

Narrative

Aircraft X was routed direct to ZZZ at 10;000 ft. which puts them through an area of high terrain of 10;200 ft. halfway through my airspace. When Aircraft X checked on I asked if they would be able to climb to 11;000 ft. for that area and the pilot responded yes. They were still in the prior sector's airspace so I told them I would climb them later. In the meantime; I was busy with weather; icing; military aircraft returning and going out to the areas; a medical priority; and two pilot deviations. None of those are valid excuses. I highlighted the Aircraft X and just got distracted enough that the flashing MSAW wasn't even enough to get my attention. Aircraft X reminded me of the climb to 11;000 ft. but it was already too late when I issued the climb.Earlier on in the session my supervisor asked if I wanted a D-side but we were short staffed and I thought I could work through it. I knew that I was busy and getting distracted with a lot of different situations. I should I have reached out right away to the prior sector and coordinated a climb before I forgot. If our staffing was better I would have accepted help from a D-side and they probably would have helped to see the flashing MSAW when I was distracted.