B737-800 flight crew reported during cruise the number 1 engine oil quantity dropped to zero. Flight crew diverted and landed safely.

Narrative

While en route at FL390 from ZZZ to ZZZ1. The secondary engine indication on the lower display popped up showing the number 1 engine oil quantity at 3. So we went ahead and accomplished the QRH procedure. Which recommends that as long as the oil pressure and temperature are within limits to continue normally. And soon after the qty dropped to 0. As we kept monitoring the situation we [advised ATC] and requested to reroute over land instead over the gulf. While the Captain is communicating the situation to dispatch. I; (FO;PF) noticed that the oil pressure dropped to the amber line and soon after went back up to about 33psi. and the number 1 engine indication on the primary engine display flashed amber. At this time we decided that it was a better idea to divert to ZZZ2.After talking to company; flight attendants and passengers. We executed an uneventful landing in ZZZ2.I think the procedure in place to blank the lower display after engine start are the best way to mitigate any excedance which make the lower display popped and therefore get the attention from any pilots.