What is the best Arduino board that will communicate with an underwater ROV via a communications protocol

There is still a lot needed to know. The most important of which is how long will the cable be. Also, will the cable need to be able to support the weight of the vehicle (haul it on board). My 1st impression would be to use RS-232. That would be good to 75 feet, officially. Rs-232 can be made to work at further distance if you use the right cable (I did this years ago). The best solution is probably going to be ethernet, but it's a little more complicated. I found this site for cable lengths Cable Distance Limits | Data | ShowMeCables.com

@sj_h1 Find out what the ROV's used by scientists and navy use. My fuzzy old memory tells me they can work at depths over several thousand feet but maybe I am misremembering. If you use fibre optic you are pretty much unlimited. Next challenge is water proofing all electronics and connections.

Interesting project, I’m also thinking there’s still a lot of unknown info tho that would help it’s success. A key would be what’s your budget?  How valuable is the ROV and how much physical space and battery power is available to you within it to add this mod?  If your sponsor can spend say $20k or more on a couple prototypes.. And, you’re not going to need to physically support the ROV’s weight, or act as a tow line tethering or powering the ROV at all.. then I’d look into using fiber optic cable. It’s often used to replace Ethernet and long RS232-like lines in military and industrial applications where high data rates for high res video and other high b/w network data comms, light weight, high flexibility and a very tough jacket to protect the cable during installation are all very important. But if you need to feed any AC or DC power to the ROV or act as a tether to support it.. fiber optics alone won’t work. You mention a need to control relays and the joystick.. relays often need some current to pass to enable a coil that flips relay contacts. Joysticks can carry small voltages that are subject to RF /EMF noise in long cables.. how long must the replacement cable be, how many conductors at what gauge are in it now and how much current must each handle? Whats the outer diameter, weight and bend radius limitations your working within? What’s the data rate of the video? If it’s HDMI or 4K, then you’ll be severely limited in distance to perhaps 50 to 100 feet max. If you Only needed to steer a ROV while it’s under say 50 to 100 feet  a sonar transducer and two way control link to a small transducer mounted into the ROV might be considered but that alone won’t support the video signal at the data rates I suspect you’ll need. 

I’m sorry I did not answer your question but these other parameters are pretty important to give you a decent reply. An Arduino Uno or ESP32 could work to control it across a sonar link if don’t need much distance or a near real time video signal from the ROV but it sounds like need to pass one and control some relays in the ROV and do it on a tight budget, some of these questions I’ve asked are germane and key to your project’s success.  Good luck .. I’ve seen a couple new and relatively cheap but capable under water ROV exploration products are now out there .. I’ll send a couple links .. perhaps they’ll give some insight.. 

Looks like for about $1200 you could get one of these Under water ROV drones. If I were doing it I’d swap the entire ROV .. or just strip out the control/ 4K video cable and controller out of one of these and mount that in your ROV .. take a look at Dustin’s YT channel .. he has several Uwater ROV reviews posted there.