Lightest weight option to provide short bursts of power (250W x 4 seconds)
Compressed air. Weighs nothing. Can be put into fibreglass/carbon tubes inside the board. Could be recharged from a dive tank (with pressure reducer), or a small 12V compressor and solar panel.

You can probably make a single pulse jet where there is a tube inside the board with a jet nozzle at the back (and flap at the front) that fills with water. 

When you gate the compressed air into it, it shoots the (single) water charge out the back bottle rocket style. A 2m x .1m sq tube holds 20kg of water, and can give a big impulse to an 80kg rider+board. 

More likely you would use a small tube as a pulse jet and just put a series of bursts of air into it. Once each charge is ejected and the pressure drops, the flap at the front opens, and it refills with water ready for the next burst.

You are clearly able to work out the energy/mass/pressure requirements of such a system yourself

Another alternative is butane cartridges  a la Pasload nail guns. But you do have to keep the igniter dry.



From Wikipedias 530kJ/5L@200bar I estimate about 3kJ/l of air at 100psi, in isothermal exansion. That might be possible if the air is forced through the water to regain its heat on expansion. 

So 7 drink bottles for your energy needs. Plastic drink bottles are good for about 100psi, probably more wrapped with fibreglass tape (which stops the spherical bulging).

There are also carbon fibre 300bar tanks 0.5L/600g paintball tanks giving you over 50kJ and 3l/2kg scuba tanks for 350kJ