Air Carrier Flight Crew Flying 737NG aircraft reported Stabilizer Trim malfunction inflight.

Narrative

Passing FL250; while climbing out from ZZZ1; the CA (Captain) (PF (Pilot Flying)) called my attention to a STAB OUT OF TRIM light. We requested a level off from ATC and the CA asked me to run the Non-Normal 700/800/900/ER Stabilizer Out of Trim checklist. Following each step; we confirmed the stabilizer was not trimming with the autopilot engaged; the CA turned off the auto pilot/throttles; we confirmed the stabilizer was not properly responding to manual electric trim inputs (i.e. only nose down trim inputs were able to be applied electrically); and; we took note the last step directed us to execute the Stabilizer Trim Inoperative procedure next. At that point; the CA asked me to contact the Flight Attendants; give them a synopsis of our situation; and request they bring the Company 737 CA up to the cockpit to render assistance. Upon entering the flight deck; the CA briefed the DH (Deadhead) on the situation; he verified our assessment of the aircraft malfunction; and the CA asked him to back us up. Then; I read and accomplished the 700/800/900 Stabilizer Trim Inoperative Checklist (up to the Deferred Items section) while the CA and DH confirmed each step. Next; the CA transferred aircraft control to me; gave the flight attendants a briefing; coordinated with Dispatch; and solicited inputs regarding a potential divert from the other flight deck members. With the support of Dispatch and the cockpit; the CA elected to divert to ZZZ; notified ATC and Dispatch; [advised] ATC (due to a higher work load associated with manual trim requirements); and; coordinated for us to land on the longest available ZZZ Runway (XXR). Further along our route; the CA calculated our VREF15 approach/target speeds and Non-Normal landing distance; briefed the arrival and Flaps 15 landing; and; resumed PF duties. Shortly thereafter; we performed the Deferred Items checklist; flew an uneventful arrival; accomplished the landing checklist; and the CA executed a smooth FL 15 overweight landing. Afterwards; we taxied normally to the gate; shut down the aircraft; submitted maintenance write ups for the malfunctioning stabilizer trim and overweight landing; conducted a face-to-face briefing with Maintenance personnel; and completed the Parking and Termination checklists. A couple hours later; we flew the passengers on a different aircraft to their intended destination (ZZZ2) without incident.