I use a 10AH Sealed Lead Acid (SLA) battery. My HB 597ci HD DI draws 615mA. Thus I should get roughly 16 hours out of my battery. Dividing the AH by the draw is fairly accurate here since the current draw is quite low. In the long days of summer I am sometimes out there for 12-14 hours so this size battery suits me well. I am not too worried about the weight. I weight 215lbs so even a "heavy" battery is a small increase in the overall load of the kayak.
To decide what size battery you want, you can find out how much current draw you need to support and how long you will be out. Fewer AH is less $$$. The other nice thing about an SLA is that is has a gradual discharge curve. I can check the voltage on my battery (from my FF) and know if I am getting low. Li-ion batteries are near full voltage right up until they fail. This is great for power tools, but I think less well suited for a FF application where they can operate at lower voltages before ceasing to work.
If you want to save weight get a Li-ion battery. If you want a less expensive and more robust option then I would recommend an SLA battery.
Fish tremble when they hear my name
A ship in harbor is safe -- but that is not what ships are built for.
--John A. Shedd, Salt from My Attic, 1928