According to this article:
Tom Szentgyorgy, an executive producer for The Mentalist, said he and other producers mulled several cities for Jane’s new locale before settling on Austin.
“We were looking for a city outside California where Patrick Jane could start over. We wanted it to be a city with a distinctive character, a city with some funk to it,” Szentgyorgy said. “At the same time, we wanted it to be a city whose architecture and look weren’t too familiar to television audiences — because, frankly, we continue to shoot our exterior scenes here in Southern California.”
In addition, he said, Jane’s new home base needed to be a city where it made sense for the FBI to maintain a significant presence. (Note: The fictional FBI office in Austin looks much fancier and bigger than the real one.)
This likely originated from the earlier storyline:
Another team member, Kimball Cho (Tim Kang), is in Austin, the previous stomping ground of FBI agent Dennis Abbott (Rockmond Dunbar), who is assigned to close the CBI after its infiltration by a secret society of corrupt California law enforcement agents, including the man at the center of the Red John manhunt, CBI director Gale Bertram (Michael Gaston).