Southwest Airlines flight 3508 encountered unanticipated moderate turbulence while deviating around convective weather. The flight originated from Dallas Love Field Airport (DAL), Dallas, Texas, with a planned destination of Northwest Florida Beaches International Airport (ECP), Panama City, Florida, and diverted to Jacksonville International Airport (JAX), Jacksonville, Florida. Preflight weather planning indicated scattered convective buildups in the Florida panhandle region, typical for that time of year. The crew designated JAX as the diversion airport. The flight departed DAL uneventfully, and en route operations were normal with the seat belt sign off and cabin service underway. As the flight descended toward ECP, the flight crew received updated weather information showing developing convective activity near the airport. Air traffic control (ATC) vectors corresponded with what the crew saw via enroute weather monitoring. Another Southwest Airlines flight ahead of them executed a missed approach at ECP due to heavy precipitation and reduced visibility. The flight crew determined that available fuel on board would not permit multiple missed approaches and a diversion, so they elected to divert to JAX. According to the flight attendants (FAs), the seat belt sign was illuminated, and the flight crew reminded passengers to remain seated. However, a passenger disregarded the seat belt sign and entered the aft lavatory. During the climb to the east, through approximately 5,000 ft mean sea level in instrument meteorological conditions, the airplane encountered a brief period of moderate turbulence lasting 2 to 3 seconds. The autopilot disengaged, and the first officer assumed manual control. The flight crew stabilized the airplane and continued the diversion to JAX. When the airplane encountered turbulence, the passenger was exiting the aft lavatory and was thrown upward, striking the ceiling and the floor. The aft FA reported the passenger lost consciousness briefly before regaining awareness and sitting in the nearest available seat. The aft FA notified the flight crew of the injured passenger, and emergency medical personnel were requested to meet the airplane on arrival. The flight landed uneventfully at JAX. Paramedics met the airplane and transported the injured passenger to a local hospital, where the passenger was diagnosed with multiple fractures to the neck and back and subsequently underwent surgery. No other injuries were reported and there was no damage to the airplane.