The A7 III, with its compact and ergonomic design, has been a very popular camera for video recording. The battery life is drastically improved compared to the previous A7 Series cameras, which was a primary complaint of Sony Alpha users and is particularly beneficial for underwater video, as it allows you to shoot for much longer without the need to open the housing. The A7 III also has dual SD card slots, which play an additional role in the maximum allotted time you can record underwater without having to open the housing. The auto-focus during video works very well.
You'll also want to keep your specific needs in mind. Not all cameras are suitable for every sport, and certain form factors lend themselves better to certain activities. On top of that, different shapes allow for different mounting accessories and possibilities. If you want to catch a unique perspective, like an under-skateboard shot, you'll want to pay close attention to size.
The Nauticam Panasonic S1/S1R housing offers an M28 bulkhead, designed to be used with the HDMI 2.0 cable with the Atomos Ninja V monitor/recorder. When shooting the Panasonic S1 with the Atomos Ninja V monitor/recorder, you will immediately be able to record 10-bit 4:2:2 HDR footage direct to Apple ProRes or Avid DNx. This is highly advantageous and the Nauticam housing for both the Panasonic S1 and the Atomos Ninja V work hand in hand to provide a particularly user friendly system.
Then there's resolution and video quality. You'll want a camera that can record in 4K for sure. The top-end models from DJI and GoPro offer 60fps 4K with digital stabilization that's absolutely incredible. They're also not limited to ultra-wide views; you can set a narrower angle to remove lens distortion, making them solid choices for vlogging and travel logs.