Arduino powered relay behaving oddly
It seems that:


when the LED dims, current is flowing through the coil and then through the LED. Because the coil allow low current to flow through it, you see low brightness into your LED.
when the relay clicks again an high voltage is produced on the coil and the LED, then you see an high brightness in your LED. This is interesting if you are a novice because you can see your LED doing the function that, in the original circuit, would be performed by flyback diode.


Solution:


flip your relay left-right