Which arduino board would you recommend?

Wow! Look at the corrosion/sludge in there. You'll need to protect your circuitry from exposure (potted?) but also provide for movable parts. I'm thinking a low power motor, attached to a long screw. Everything can be inside a protective capsule, with only a small, o-ring sealed opening for the screw. Attach your foam ball to the screw, so it moves up and down, depending on which way the motor turns. I'd start with your biggest LiPo for power.