ZZZ ARTCC Controller reported a C560 deviated from clearance entering a higher MVA resulted in a CFTT event.

Narrative

As the R-Side Controller on Sector XX/XY I had a multitude of aircraft deviating South from Sector XZ around weather; creating an influx of traffic volume/complexity. The Controller training on my D-Side took action to shift over to Sector XY to open it to alleviate the traffic volume. During the briefing to split the sector; ZZZ Center Sector XA pointed out Aircraft X descending into ZZZ. I was unable to hear them calling over the landline due to the traffic volume and briefing taking place. Once I split the sector and was attempting to catch up on Sector XX by itself; I noticed Aircraft X was in handoff mode; so I accepted the handoff. The aircraft checked in almost immediately; saying they had been cleared for the RNAV Runway XX approach into ZZZ; but did not think they could make it in due to the weather on the approach path. I then called the weather to them; and gave them a clearance back to ZZZZZ to hold as published at 110. I got pulled away to take care of other tasks; and when I came back to Aircraft X in my scan they were off-route heading North at 110 into a 120 Minimum Instrument Altitude (MIA) block in ZZZ1 Center sector XA. They said they couldn't hold over ZZZZZ due to the weather and needed to go to ZZZZZ1 to hold. I issued a low-altitude alert; and climbed Aircraft X to 130 for the high terrain blocks. I then called ZZZ1 Center Sector XA to coordinate the situation and request control. Aircraft X then said they wanted to divert to ZZZ2. Again; I called ZZZ1 Center sector XA to coordinate. I cleared Aircraft X to ZZZ2 at 130. A minute or two later Aircraft X again called; requesting to divert to ZZZ3 this time - requiring another call to ZZZ1 Center Sector XA as they were still about 5mi inside of their airspace. At that time my D-Side called ZZZ1 Center Sector XA to coordinate the diversion; and all was taken care of. One thing that may have prevented this situation; is a more proactive split of Sector XY from Sector XX to alleviate the traffic volume; combined with the complexity of Sector XX. Another issue that occurred was some frequency issues due to there being weather between the transmitter and the aircraft needing control instructions.