B737 Captain reported low oil pressure indication led to engine shutdown and divert.

Narrative

At cruise we had low oil pressure annunciate with red band oil pressure on the number one engine. QRH procedures were followed to an engine shutdown with a subsequent divert to ZZZ.At the time we were going by ZZZ off to our left and could see the field clearly. The First Officer (FO) was the pilot flying (PF) and I the pilot not flying (PNF). We confirmed the problem to make sure we were doing the correct checklist and I immediately [requested priority handling] with ATC and we descended to the engine out altitude for the aircraft. In the descent we went through the Engine LOW PRESSURE checklist. This directed us to the Engine Failure or Shutdown checklist which was run next. After confirming the checklist was appropriate I completed the checklist to step 15 where it directed to go on to the one engine inoperative landing checklist.At this time and also during the checklist I kept cross checking with the FO and contacted dispatch to confirm ZZZ was a good divert field. The FO kept flying and descending to ZZZ. I briefed the Flight Attendants with the briefing and told them to plan for a normal landing. Dispatch ensured that emergency equipment would be available. I made a PA to inform the passengers that we had an engine problem and had to shut it down and would be landing in ZZZ. ATC gave us about 5-10 minutes of delay vectors as I went through the checklists.The One Engine Inoperative Landing checklist was completed to deferred items where the FO and I came together again to brief the ILS approach to XR. The deferred items were completed and when we were ready the FO flew a nice single engine approach and landing to XR. We cleared the runway and had emergency vehicles go over the plane for any possible hot brakes or fire and then taxied to the gate.Things that could have gone better:During the descent when I was running the checklist we passed FL180 and the FO made the call for altimeters set. I set the altimeters but did not turn on the landing lights. It was such a nice night and ZZZ airport and XR was so bright it didn't even occur to me until the landing roll that they were off. It was a mistake on my part and the workload I was under at the time contributed to the mistake.I also did not give a lot of consideration to the fact that we were going to land at max structural weight or over by a few hundred pounds. I should have briefed the FO that we would be. He made a nice landing and Maintenance asked me after the flight to add in that we made a normal landing in the Maintenance write up. I was more concerned about getting the aircraft on the ground safely since we had shut down an engine.