Allegiant Air flight 485 flight crew performed a climb maneuver in response to a Traffic Collision Avoidance System (TCAS) advisory resulting in a serious injury to a flight attendant onboard. Flight 485 was at flight level (FL) 230 [23,000 feet] and enroute to the Blue Grass Airport (LEX), Lexington, Kentucky when the event occurred. The flight crew reported they had been on a generally northern course since their departure from Fort Lauderdale-Hollywood International Airport (FLL) when they received a heading of 100 degrees from air traffic control (ATC). Jacksonville Center closed the airspace to the North in response to severe weather in the area. Minutes later, the flight crew reported receiving a TCAS traffic advisory (TA) followed immediately by a resolution advisory (RA) to which the flight crew followed the given climb rate, to avoid another aircraft in their path. After the flight crew returned to FL230, they received a call from the lead flight attendant to report that the unexpected climb caused one flight attendant to be “knocked to the ground” and was injured. ATC declared an emergency for the flight after the flight crew reported the injury of the flight attendant and the flight subsequently diverted back to FLL due to significant weather over central Florida.