B737-800 First Officer reported performing an inflight engine shutdown due to low oil pressure. The flight crew then diverted to an alternate airport.

Narrative

Takeoff from ZZZ was uneventful. On climb-out; around 10;000 ft. the mini-engine display appeared and the oil quantity indicator on the right engine began slowly counting down from around 80. During the oil quantity decline; there were no other indications of an engine event; other than the quantity being highlighted white. No oil pressure light or indication; oil temp or vibration coming from the engine. We continued the climb to our final cruise of 19;000 ft. and reviewed the QRH in preparation for the event that something might happen. The QRH references that the LOW OIL PRESSURE light may or may not be illuminated. During the whole event we did not see the OIL PRESSURE light illuminate. We ran the one step for ENGINE LOW OIL PRESSURE which refers you to the Engine Failure Checklist if the pressure is at or below red line. The pressure was still within limits - green - and had not declined at all from the comparable left engine pressure.The quantity continued to count down and when it got to around 20%; we asked for priority handling to ZZZ1; a shortcut; but we did not declare an emergency. We were given direct ZZZZZ for the RNAV XXL or somewhere thereabouts and we were given a descent. It was around then that the oil quantity indicator reached 0%; still with no other indications. The QRH was on the dashboard; opened up and ready to go. During the initial descent; the oil quantity actually counted back up to 20% and fluctuated up and down. We got the ATIS and landing data for ZZZ1 Runway XXL and briefed a Flaps 40 visual landing. The landing distance available for Flaps 40 was around 1;400 ft. with brakes MAX; wet runway. It was very visual conditions in ZZZ1 with a light wind out of the west and a 5-WET GOOD field report for Runway XXL. We briefed the arrival and approach and ran the Descent and Approach Checklists. It was around this time that the oil pressure indicator on the mini display rapidly went to zero. The mini display rapidly flashed white; then amber; then red and the engine temperature began to climb on the mini display. Captain called for the Engine Shutdown Checklist and [asked for assistance] with ZZZ1 Approach. The Shutdown Checklist went uneventfully. We were below 10;000 ft. at this time and we may have exceeded the 250-kt. speed limit; but neither of us could recall by what small margin and for how long. We had five hours of fuel onboard - we were tankering 15;000 lb. We briefed the flight attendants and made an announcement to the passengers.By this time; we were on the approach for [Runway] XXL and had ZZZ1 in sight with a single-engine configuration. We ran the landing data for a single-engine landing on [Runway] XXL and noticed that the landing distance available was an acceptable margin with the very favorable VFR conditions. We discussed diverting to ZZZ2 and agreed upon continuing the approach to ZZZ1; with the plan of executing an early missed approach to [Runway] XXL if we were uncomfortable or outside the parameters of a stable approach with the plan of immediately diverting to ZZZ2. The approach; final approach; and landing were normal. We touched down with brakes MAX and exited Runway XXL at our normal exit for this runway. We taxied to the Ramp and had Crash Fire and Rescue inspect the engine for heat or damage - there was none - and continued in to park at the gate.In conclusion; I am proud of the way this event played out. I was comfortable with our plan to continue the approach considering both my experience and Captain's; not only with the 737 but also with flying into ZZZ1. We had discussed all options and eventualities within the short time available and I was satisfied with our Plan B" as well as a "C."This event unfolded very slow but with a very rapid conclusion. It very much reminded me of the loft where we had to decide rapidly between ZZZ3 and ZZZ4 with a fire on board. I think more training with this type of slow engine failure would be helpful."