Since it is going to be donated to a public entity, it just might be tax deductible.
What you are asking of the membership here is very complex and is not a task that is normally pursued on this Forum. It might have been best to ask if support of this kind would be available prior to purchasing the hardware. Do you have any experience with the Arduino microprocessor family and boards and can you read electrical schematics? If not, you are going to have a tough time with this project.
I would ask your local junior college microprocessor course instructor or a knowledgeable student for help. Since it is a project to be donated to a public entity you might just get some free help.
Here are some links to discussion threads on this Forum where members attempted to do what you propose to do:
Arduino Mega & BBB
Arduino Cape PCB
BBB & Arduino Cape
Arduino Uno or Mega
If you are going to use the Arduino Mega 2560 Board then you will probably need a Shield to go on top of the Mega Board to get access to the Arduino 2560 Digital I/O and Analog I/O Channel pins as shown on the OpenROV 2.8 Schematic Cover Page.
The HomePlug Adapter on the ROV end and the BBB both get their power (+5 vdc and +3.3 vdc) from the Controller Board and I doubt that the Arduino Mega 2560 Board can supply the necessary current for the HomePlug Adaper (+3.3 vdc) and the BBB (+5 vdc) without possibly going into overload.
To connect the BBB to the HomePlug Adapter on the ROV end you will have to fabricate this OpenROV RJ45 Connector. The BBB will communicate serially with the Arduino Mega 2560 over the Arduino’s primary serial UART. The BBB communicates with the Topside Adapter/Laptop through the HomePlug Adapters via the Tether.
Also, unless you plan to duplicate the current and voltage measurement circuits on the Controller Board, you are going to have to extensively modify the Controller Board firmware which will be no easy task.
I am a Systems Engineer with over 30 years experience with Digital/Analog Control Systems and there is no way that I would tackle this project when the Controller Board solution is available due to having to ensure that you connect all of the off board components to the right pins of the Arduino Mega 2560 Board and have not made any mistakes which could be disastrous. You will need a good low wattage soldering iron and a Digital Multimeter for this project.