Air Carrier Captain reported First Officer felt ill due to odor in the cockpit. Captain determined an air return was necessary. The crew requested priority handling and executed a safe overweight landing.

Narrative

As I recall: At approximately FL250 climbing to FL290 the First Officer (FO) asked if I smelled exhaust. I said I didn't. We leveled FL290; initial cruise altitude. She said she felt lightheaded and was going to take some oxygen from the mask. I think that was for roughly 2 to 3 minutes. I still didn't smell anything. When she returned; she requested that we land. I was surprised; but certainly understood if she felt that way that we would land. Coordinated with Dispatch to determine where they would prefer that we go. Their choice was ZZZ. By the time we turned around; ran checklists; coordinated with ATC; Company; briefed FAs (Flight Attendants); passengers; I figured we'd be landing at ZZZ (landing as soon as possible). That choice made sense. ATC handled everything professionally and were very accommodating to our requests. Dispatch was great; helpful... provided the landing data for numerous different runways. He also indicated that [priority handling] must be declared due to the overweight landing. When safe on deck; the Chief Pilot informed me that that was incorrect (for future knowledge). I wasn't 100% sure at the time; just thought; no harm no foul; [requested priority handling] and continued with the plan. At one point I called; informed our Flight Manager of the situation. I asked her if she smelled exhaust fumes in the cabin. She said she didn't. We asked her to come up to the cockpit to see if she smelled fumes up front... She said she did. She then returned to the cabin; I called her a few minutes later; she said she still couldn't smell exhaust in the cabin. I never smelled; felt or saw anything; but the FO did; and that's the point. The FO managed to continue her duties; even though she wasn't feeling well. She did a great job. We continued. Reviewed Smoke; Fumes checklist; Overweight Landing checklist; requested to land west instead of east as ATIS was advertising. Approved as requested. We transferred the controls and I landed the aircraft; 25R. Uneventful. Opted to taxi to the gate; Crash Fire Rescue trucks followed us in. They checked the brakes; were ready to help anyone if required. To my knowledge; no passengers reported smelling fumes; the lead FA confirmed that. Once shut down; customer service came aboard; as did Maintenance. We called Dispatch; the Chief Pilot and [I] called it a night. It won't surprise me if Maintenance finds something. I fully trust that the FO (and the lead FA) smelled something; it was most definitely making the FO nauseous. We needed to land.