Fairy lights dimming over time

There is only 6 failure points in no particular order.

1. The batteries. Especially rechargeable ones at end of life. Try new ones or good alkaline.

2. The leds. As the hours rack up, especially with high current flows, they can start to dim. As these are in parallel and we don't know the resistor value or the current involved total or individually, we can't say if they are near their end of useful life. Additionally overcurrent may cause failures in the diode doping mechanism which also causes shorts.

3. The resistor. The behavior you describe is unlikely but again without knowing the current involved we can't say. Thats a 1/4 Watt resistor I think so too high a current can lead to failure. Measure its resistance without load. Measure the voltage across it under load. Compare using ohms law and its color code.

4. The switch. With the likely current involved its unlikely, but these switches can start to corrode or oxidize due to the dc current and their resistance can go up. If its over lets say 5 ohms in resistance between terminals then it is bad.

5. The wire. Same as the switch, its resistance may have increased due to physical damage or corrosion. In that case the far end of the led string would be dimmer than the near end so unlikely to be the issue.

6. Battery terminals. Same as switch and wire. If dirty or corroded, say from battery leaks, resistance will build up. An eraser is a quick cleaning method for minor oxidation.

Replace the batteries with fresh alkaline ones.

What you describe is normal behavior for batteries that are either capable of supplying the current draw or are close to the end of their life. The "recovery" is due to polarization.