B737-800 Captain reported shutting the number 2 engine down after receiving low oil pressure indications and diverting for landing.

Narrative

In cruise flight we observed oil quantity on #2 engine drop to zero. I pulled the QRC and designated the First Officer (FO) to continue flying the aircraft. Per procedure; we waited until oil pressure to get to red line before shutting down the engine. In that time I sent an ACARS to dispatch about a diversion to ZZZZ and advised ATC. We were cleared direct to ZZZZ and descent to a lower altitude. Oil pressure was still slowly dropping so I used the remaining time to three chime the Flight Attendants and [advise them] and make a PA. Oil pressure dropped to red line and we shut down and secured the #2 engine per the QRH procedure. I then ran the numbers for an overweight single engine landing distance and briefed the FO the QRH information. Together we briefed an RNP approach and I completed all regular checklist items as well as abnormal landing QRH items. I then checked in with the Flight Attendants and briefed them for a normal landing and plan to stop on the runway with airport rescue and fire-fighting (ARFF). We were then cleared for the RNP approach with the airport in sight and I supported the FO with the task at hand. After a normal touchdown and rollout we stopped and I made the immediate PA to remain seated. After ARFF checked the aircraft I made another PA before taxiing to the ramp area and securing the aircraft.