What determines how much energy a stone can hold? Imagine you have a stone that is being hit by focused sunlight from a magnifying glass. Later you will place the stone into water to heat it. What type of stone will transfer the most energy into the water and why? 
 A: You're problem is not fully defined, but let's begin to define it. 
Firstly, you're looking for something with a high emissivity $\epsilon$, so that it will absorb most of the incident light. This factor is simple: the higher $\epsilon$, the swiftlier you can transfer heat to your stone. As long as the stone doesn't get too hot, the rate of absorption greatly outweighs emission (by the Stefan Boltzmann law).
Next comes the question of transferring the stone's heat to the water, and here is where things get complicated. You need the specify how you are going to do this. In one scheme, you could set up the stone so that it was just below the water's surface:

In this scheme, at steady state, the only mechanism for loss is where the stone is in contact with air or just below the surface local heating leads to a locally high temparature stone / water, and this can then lose heat to the air. You can make this small (1) by ensuring that most of the stone is steeped deeply in the water - so you want a long thin one and (2) making sure that the conductance (rate of heat transfer per unit area across the stone-air interface) is small compared with the stone-water conductance. If these conditions are met, then at steady state the stone will be almost at a uniform temperature and hotter than the surrounding water.
A second scheme is this: you transfer heat in cycles by heating the stone, then steeping in water to let it cool there, then taking the stone out and repeating the cycle. Here is where the problem is ill posed: what's best? Do you want to transfer heat as quickly as can be? Do you want the water's temperature to rise the fastest? If so, then three factors weigh on your scheme: (1) the stone's heat capacity $S$ (mass times specific heat capacity), (2) conductance across the stone-air interface $C_{s,\,a}$ and (3) conductance across the stone-water interface $C_{s,\,w}$. Now let the time throughout which you heat the stone be $t_H$ and the time throughout which the stone is steeped in water be $t_S$. Let the incident light's intensity (irradiance if you're an illumination specialist) be $I$ and the stone's temperature at the beginning of this cycle be $T_b$.
As the stone is heated, its temperature $T$ follows 
$${\rm d}_t T = \frac{I}{S}-\sigma \,T^4 - C_{s,\,a} (T-T_a)$$
where $T_a$ is the ambient temperature (the first term is the heating term, the second the Stefan-Boltzmann-described radiation loss and the third the conductive / convective heat loss). You can then write down another coupled differential equation for the temperature evolution with the stone steeped in the water, and set up an equation for the amount of heat transferred each cycle, which you can then use to explore your problem. I have run out of available time for now, and may come back to this problem, but this should get you going.
