Electrical – Open Hot on GFCI outlet

Best I can tell, it's time to buy another GFCI (the correct one, this time -- hopefully, the third time's the charm)

There are four problems here:

1. The old GFCI finally died (at power on, likely) and the new GFCI is likely DOA. Easiest fix is to try getting another GFCI and putting it in.
2. Your new GFCI was the wrong part -- you need a weather-resistant receptacle in a damp or wet location these days (NEC 406.9), and yours isn't. Leviton's W7599 is the equivalent of what you have in a Weather Resistant model, BTW (as of this post, at least).
3. Receptacles in wet locations require a while-in-use weatherproof cover or "bubble cover" these days, such as this. This helps keep rain out even under wind-driven conditions, no matter whether stuff's plugged in or not.
4. Even weatherproof boxes need to be pressure equalized (via a drainage hole in the bottom) in order to keep pressure differences from driving water through even the slightest imperfections in seals and to give any water that does get in a path to escape. This is allowed as per 2014 NEC 314.15:

314.15 Damp or Wet Locations. In damp or wet locations, boxes, conduit bodies, and fittings shall be placed or equipped so as to prevent moisture from or accumulating within the box, conduit body, or fitting. Boxes, conduit bodies, and fittings installed in wet locations shall be listed for use in wet locations. Approved drainage openings not larger than 6 mm (¼ in.) shall be permitted to be installed in the field in boxes or conduit bodies listed for use in damp or wet locations.

See also BSI-004: Drainage, Holes, and Moderation for a good explanation as to why even the most snugly sealed enclosures need weep holes. So, if there isn't a small hole in the bottom already, drill a ⅛" hole in the bottom of the box, as per NEMA bulletin 110 and the aforementioned NEC clause.

If rain is getting into the box from outside, you could put a new rain resistant cover on the outlet which might solve the problem.

If there is a leak into the box that you cannot immediately stop, then you can remove the receptacle and wire the circuit through with wire nuts. This will prevent this one spot from tripping a GFCI breaker or receptacle. Wire nut connections are much more resistant to moisture than a receptacle.

I had to do this for a receptacle on an outside wall which was getting wet from a roof leak into the wall and tripping an essential circuit protected by a GFCI breaker. I used regular Scotchlok (red size) wire nuts, filled them with antioxidant paste, and turned the nuts upward so they would shed water. I attached a blank cover plate over the box.