A TRACON Controller cleared an aircraft for a Localizer Approach to a runway where the localizer was set to the opposite runway and the aircraft flew below the Minimum Vectoring Altitude. The airport's Control Tower was closed for the night.

Narrative

ZZZ Tower is closed; they set the localizer to Runway XXR. Winds were strong from the west. Aircraft X checked in going to ZZZ IFR. I asked him to verify he had the weather and which approach he'd like. He confirmed he had the weather and initially he responded with the RNAV Y [Runway] XXL; which was available; so I continued him inbound for the approach and used my data block entries that reminded me I had everything I needed to clear him for the approach. He later stated he was unable to do the RNAV and needed the Localizer to XYL; which was unavailable. I failed to recognize the LOC XYL was unavailable. I cleared him for the LOC XYL and after he appeared established on the course; changed him to advisories. After about 2 miles; he appeared to be slightly off course; I gave a low altitude alert; in the chance he was still on frequency. I'm not sure if he received it; but very shortly after checked back in executing a go around. He attempted the approach one more time with a similar result. He ended up diverting back to ZZZ1 and landing there.If the procedure is for ZZZ to set the localizer to Runway XYR at closing even when the winds strongly favor the runway XY's; we could just have the LOC XYL NOTAM'd out of service for the hours the Tower is not in operation. That way the flight crew would know of the situation. As it stands now; this operation has a single point of failure in the controller working the ZZZ approaches.