For a vehicle, what is the most efficient way to store energy from various inputs?
More later maybe. Quick thoughts:

Various electrical inputs via MPPT (maximum power point tracking) converter to LiFePO4 (Lithium Ferro Phosphate) battery is usually likely to be the most efficient and cost effective way of storing electrical energy. See end for MPPT comment.

Brushless DC motor drive is a top contender for drive efficiency with regeneration of "braking energy" back into the battery where approriate. 

Stirling engines are great but are unlikely to be practical from cost or mass density or volume density points of view in real world solutions. 

Energy conversion from low grade heat sources such as temperature differentials is extremely inefficient due to Carnot efficiency limit of (delta temperature)/(Maximum temperature) with actual efficincies being a fractin of that. Very low % :-(. Such may be OK for static applications where getting free energy is much more important than weight or cost or size.)

Fuel cells have their place but Hydrogen is hard to deal with well and the technology to use it compactly is still evolving. It's very low mass density and high diffusion rates and other factors make it an unlikely solution in compact portable storage and powering applications. Methanol cells can have higher energy densities but are not yet good as storage solutions.

LiFePO4 batteries can store energy at > 90% efficiency have very good but not superb density compared to th very best battery technologies and have good life-cycle costs (but higher initial costs than eg lead acid.) Lead acid can be extremely good on conversion efficiency with care and has lower initial costs but higher long term than LiFePO4. Various other LiIon storage systems are not as good as LiFePO4 energy wise but have higher energy storage densities. 

"Just paddling" has its place but can be over-rated :-).



@Rocketmagnet's suggestion of a sail is even better than he suggests. A practical sail for a Kayak can be of modest size and can be highly practical and provide a very good motive source on lakes and in the sea. You may need Exalted-Grand-Master status to use it above a class 2 rapid - but that may apply to trolling motors as well ;-).

Rocketmagnet's suggestion of using flowing water as an energy source when stopped is also a good one (and is related to my braking energy and regeneration comment). Total potential energy in falling water is mgh = mass x gravity constant x head ~= 10 Watts per kilogram.meter/second or about 12 Watts per gallon.foot/second. Extractable energy is probably around 10% of this  in a portable propellor situation.
 Of more likely interest is energy due to velocity = 0.5 x m x V^2.
  Below -
 density = kg/m^3
 V = metres/second. 1 m/s ~= 2.2 mph,
 Area = prop area in metre^2

Mass/second = Area of prop x velocity x density so
 Power at 100% conversion =~ 500 x V^3 x A    Watts. V in metres/second. 
 60% of this max unducted.
 Near 100% of this in superb design ducted.
 Say 20% or so in real world. So
 Power ~= 100 x A x V^3 Watts.  

Interest only: The above formula also works for wind turbines with a factor of  about 1000 x less power per area for air due to the density difference.

Carrying a small wind turbine with fold out blades for use when stopped can make sense. Use when in motion in a kayak again needs Grand Master status. 



MPPT

Wikipedia on MPPT

They say:


Maximum power point tracking (MPPT) is a technique that grid tie inverters, solar battery chargers and similar devices use to get the maximum possible power from one or more solar panels.[1] Solar cells have a complex relationship between solar irradiation, temperature and total resistance that produces a non-linear output efficiency known as the I-V curve. It is the purpose of the MPPT system to sample the output of the cells and apply the proper resistance (load) to obtain maximum power for any given environmental conditions.[2]


MPPT 2 page introduction useful.

Similar + product info similar

Similar

MPPT is useful with many  energy sources.
 An excellent way of thinking of it is as being an electronic gearbox that takes current and voltage at inout and outputs a different voltage and corresponding current such that 


Vout x Iout = Vin x Iin x K


where K is the efficincy of concersion.
 For a given set of operating conditions MPPT adjusts the effective load resistance and thus the voltage and current such that maximum power is being obtained and adjusts output voltage or current to suit the target output device.
 An example would be an industry standard nominally 12V crystalline silicon PV panel (= photovoltaic = solar panel) charging a 12V lead acid battery. A stanradd panel has 36 cells and an output voltage in full sun of 18V or more.  At peak power point (the MPP that MPPT tracks) the voltage will be ABOUT 15V but this varies wit cell efficincy, insolation (sunshine level), age of PV panel, cleanliness of glass, atmospheric conditions and more. The bttery may be optimally charged at a voltage of anything from about 10V (rather dad battery) through 14V+ (certain specialist modes). The MPPT controller matches these different voltage and current levels. If the battery was best charged at 12V and the PV panel MPP was at 15V then if the two ar just joined together the efficiency of PV energy use is 12V/15V = 80%. The 20% extra is lost. This does not mean that the battery necessarily uses the optimum energy as well as it should but the problem then changes from a PV panel loading one to a battery chemistry one.



LiFePO4:

Long cycle life, good temperature range, superb current charging efficiency, very good to excellent energy charging efficiency, relatively robust. relatively flat voltage range, excellent high temperature performance acceptable to good low temperature performance, lowest whole of life cycle-cost of any battery.

This applies t