B737 First Officer reported a loss of a hydraulic system and diverting to a nearby airport.

Narrative

This was the airplane's first leg in multiple days. The Maintenance Release was still broken as we were on the way to the airport with multiple open items. The new Maintenance Release was received just prior to push back. During the climb phase; a caution light illuminated for the low pressure of the ENG 1 HYD EDP. We leveled off and followed the QRH procedure and consulted Maintenance Control via VHF radio to get further information. Upon review of the systems page showing stable and adequate quantity and pressure; and with guidance from Maintenance; we elected to continue the climb to cruise. Roughly 10 minutes after reaching cruise phase; it was noticed that the HYD quantity in A had dropped to the standpipe levels of approximately 18 to 20%. We contacted Maintenance Control via voice for guidance and were told to use the Loss of System A procedure in the QRH to continue as we expected. With agreement from Maintenance Control and Dispatch on that call; the airplane was diverted. Shortly after this; we reviewed the QRH procedure and decided to [Request Priority Handling] for a myriad of reasons including; but not limited to; reliance on single/alternate/standby systems and the handling requests from ATC that we desired; such as a long final and Runway XXR. We provided the briefing to the FA's shortly after turning around and made the appropriate PA's. From this point; descended early to burn as much fuel as possible for the landing; yet having to stay above some bands of rain. Followed the Loss of System A QRH and landed uneventfully on Runway XXR and taxied to the gate without incident. Landing was overweight by approximately 1700 lbs with a normal touchdown and roll out.