Every few inches there's a little cup that fills with water as the rain from the gutter pours down the chain. When it's full, it tips over and dings a little bell mounted next to it.
The water then goes to the next cup and the process is repeated, but the lower bell is slightly larger than the one
above it and plays a lower note.
The effect is a sound that matches falling water. (link)
Might just have a specific unit you hang from a tree with a funnel up top, but not sure you'd get enough water. Cups allso might not be necessary, paddles would probably work fine.
Could also just have the chain links get bigger so the sound gets lower and have them get struck by the simple paddle things. It'd be quieter but that might be okay.
Actually, I think just selling a bunch of progressively bigger bells that you hang on the chain would work fine, use the chain as the striker. Might have a little element sticking out from the bell itself into the water stream.
Might not even need bells, little wind chimey like metal tubes could work too. With that you could make the whole device both the rain director manager and musical thing at once.
Make it small enough you could even just lower it into a rain spout pipe, get some resonance added to the sound.
Bells floating in a cup might work too.