With the example that you gave above, I would use a chunk of lead. The LED module needs to be attached to the heat sink so whatever the LED is in, the heat sink needs to be in as well. If you want to keep the unit small like using a SSC enclosure, I would go with mass over size (surface area) to give it the ability to absorb and store more heat.
Lead!? Um, no. Gravitational mass and thermal mass are two different things, and not always closely linked. The amount of heat required to raise the temperature of a unit mass of a material one degree C is called the specific heat of the material. Lead does not have a very high specific heat, meaning it only takes a relatively small amount of heat to raise its temperature a given amount. The specific heat of Lead is 0.13 KiloJoules / Kilogram * degrees K. (The same as gold, as it happens.) Copper is 0.39 and Aluminum is 0.91. Given its low cost and the ease of working with it, as well as its low corrosivity in dry air, Aluminum is just about the best choice for a heat sink if you want the material just to heat and cool slowly. There are a number of metals that are much better, but cost much more and are harder to work and maintain:
Beryllium - 1.83
Lithium - 3.57
Sodium - 1.21
Iron and Steel aren't too bad, running around 0.5.
For a heat sink, however, one wants not only a large thermal mass, but also a low heat conductivity. If the heat conductivity of a material is high, then even though it may have a high thermal mass, it may not tend to stay cool near the component, which is what is important. The temperature near the component will rise quickly, while the rest of the mass remains cool for a much longer time. Not only is the conductivity important, but also the total conductance, assuming the heat sink is not a closed system. In general, the heat sink acts as both a buffer and any or all of a radiative, conductive, or convective transport to an infinite medium such as air, water, or space. To this end, in general the smaller the distance from the heat source to the external medium and the greater the area of the sink in contact with both the source and the medium, the better. That is why your car radiator and your Air Conditional condenser coil is made of either thin copper or aluminum tubes attached to a myriad of very thin copper or aluminum plates. It gives the shortest possible distance between the coolant fluid and the surrounding air / space combined with the highest practical surface area to both radiate and conduct heat into the outside environment.
Aluminum has a fairly high thermal conductivity given its price - around 68 Watts / meter * degree K. Copper is much better at around 129, but copper is heavier, more expensive, and harder to work - with the exception of welding, brazing, and soldering, which is very easy with copper. Copper also tends to corrode more in dry air than aluminum. These facts are why one sees many PC cooling systems employ thin Aluminum fins as radiators along with copper heat pipes to transport the heat from the device contact plate (also usually copper) to the radiators.
Lead is very poor, at only 11.6. Even some Stainless Steel alloys (which are very poor heat conductors) are better than Lead. Indeed, there are not a whole lot of pure metals that are worse than Lead. The very best conductor of heat of all pure metals is Silver, with a heat conductivity of 136 W/(m x K). Gold is next, at 106. At this point one might notice the relative heat conductivity of the various metals and alloys falls fairly closely in line with the relative electrical conductivity. There is a reason for this. The mechanisms which produce thermal conductivity and electrical conductivity in metals have a number of fundamental processes in common, so the better a conductor of heat the metal or metal alloy is, the better a conductor of electricity it is also in general going to be, with Silver and then Gold being the best of both. Copper is very high up in the list, Aluminum is fairly high up the list, and Lead is at the bottom. OTOH, the mechanisms responsible for determining the gravitational density or thermal heat capacity of a metal have almost nothing in common with those responsible for conductivity.
Oh, and Lead is fairly expensive these days.
