Best Antenna for Groove to be Used as WiFi Booster

I bought a Groove Metal (G-52SHPacn Metal) to use as a WiFi extender on my sailboat. It will be installed on the mast, configured as CPE/router and connected via Ethernet to a router/AP in the salon. I hope to be able to access WiFi hotspots on shore while being at anchor up to 1/5 to 1 mile from them. Is there a better alternative to the included with Groove 2db omnidirectional antenna? The boat moves around at anchor so the antenna must be omnidirectional. It will be located 15 feet above the water level. Any advice would be greatly appreciated. Thanks.

Antenna gain directly depends on beamwidth. As the boat rotate (yaw,pitch,roll) your antenna must have enough beamwidth to cover all possible angles, otherwise it will drop your connection.

You correctly decided that you need omnidirectional antenna because boat moves around anchor. (i.e. yaw is 0-360 degrees)
However, you also need to account for pitch and roll rotation. I am not a sailor, neither I have knowledge of your boat - so you have to decide how much does it rotate around these axes. Based on that, you can look for antenna with corresponding beamwidth.

Second thing to keep in mind is signal strength - if your boat rotate a lot, you need high beamwidth which comes at expense of gain. This might cause an issue because you plan to anchor your boat up to 1 mile from AP. That is actually significant distance. Quick calculation:
Free Space Loss on that distance will be 104dB on 2.4GHz band and 111dB on 5.8GHz band.
Your Groove has maximum power 31dBm (on slowest speed) for both bands and included antenna should be 6dBi (couldn’t find precise specs)
(I don’t count antenna gain of access point on purpose, because firstly, it will be always different and secondly it depends how oriented such antenna is.)
Based on those inputs, on shore you will get max 31+6-104= -67dBm for 2.4GHz or 31+6-111= -74dBm for 5.8GHz

These are sufficient values to keep the link, but it is maximum possible value in ideal world. We live in real world so there will be interference (other AP in the area), reflections, obstacles (even water droplets from tide can attenuate signal a lot).
Also keep in mind that the antenna gain gets lower with each degree from its maximum point. Look for example at this chart for 6dBi omnidirectional antenna:

This is not exactly your antenna, but it will be close enough. You can see that it reach maximum gain in every horizontal angle, but only in few degrees of vertical angles. For example if your boat rolls 15 degrees from ideal angle, antenna will lose around 3dB of its gain. Personally, I wouldn’t be surprised if the boat rolls 30 degrees

Due to that, I am afraid there is not much chance to get stable connection, unless there is AP with sector/directional antenna on shore.

The antenna included with my Groove 52 is a 6dBi / 8dBi omni antenna. When a boat is on anchor in a protected anchorage it does not roll or pitch that much, though, so I think your calculations (and thanks so much for doing them for me!) are right on spot. Another question would be how well the metal Groove and antenna would stand to the marine environment. I plan on putting a lot of dielectric grease on the antenna contacts and on the Ethernet connectors before assembling. Any drawbacks to drenching contacts in dielectric grease?

Thanks again.