Looks like a standard SPST relay but the shape of it is odd. If it costs more than a normal relay I would just replace it with a normal relay and socket of the same rating which is pretty standard for an automotive relay. You could run 12V directly to the pin in the socket (87 or 30) that feeds the light and it should come on. This will tell you it's not the light chattering but the relay.

 

The first thing I would do is take 2 lengths of wire and run directly from battery to light. Verify that the light is functional. After you verify the light itself is dead or not and assuming the light is still functional, then grab your volt meter and start back tracking at each connection starting at the connection from the light pigtails, move to relay output, move to relay input. Just work back until you find the break point between having power and not, or having ground and not. Whatever is in between the break point, is failed, be it a solder joint or a component.