Trying to put figures.
One raindrop falls at 5m/s (reaching equilibrium in 1.3m) and weighs 4*10-6kg. Flashes from individual kinetic energy would be extremely weak: 170nJ=10mW*17µs. Visible in a dark night at best.
Then you could abandon individual flashes and light continuously if it rains. A strong rainfall, say 10mm in 1h, provides 35mW/m2, enough for one Led per m2. Harvesting the kinetic energy is difficult; ETFE is ferroelectric. An expensive and perfectly dry plate might convert some kinetic energy to high-voltage electricity.
The drain pipe looks easier. With 3m drop, a 3m*5m roof provides 1.2W, which a watermill can convert to low voltage while resisting dirt and moisture. Enough for a strong Led flashlight. Not very aesthetic: I too would prefer a flash at each waterdrop impact.
My general conclusion is that an external energy source is needed - consumable chemicals or something else.
Maybe a mesh of zinc wires hold just above a roof of brass sheet, but individual drops won't suffice. At (very variable) 2mS/m, one single 4mm2*1mm drop resists 100kohm, too much to supply a Led. As one dm2 is fully soaked as more drops arrive and take time to flow away, its 50ohm suffice to light continuously one Led. Maybe the zinc and copper can be printed side-by-side in a pattern on an insulator. With only one Cu-Zn cell, it would take an electronic circuit to provide the voltage for a Led.
My preferred solution for aesthetic is to have a transparent roof sheet (glass, Pmma, polycarbonate) where you inject light at one edge and absorb it at the others. With proper design, light stays completely trapped in the dry transparent sheet, but every individual drop permits light to escape the sheet to the drop and they the air. That would be the best visual effect. You need the sheet to dry and clean spontaneously (maybe FEP, but it costs). Led are better to save electricity, choose the colour and inject light at a good angle, that is, near to escape the sheet. Injecting light from several edges may keep a uniform effect in strong rain, and ambiant light detection can adapt the intensity (PWM) to day and night.
At proper angle, the edges too can reflect light to recycle it and save electricity. Less nice under heavy rain.
I wouldn't supply the light by the drainpipe watermill, because I like to see the very first drops on the roof.
Marc Schaefer, aka Enthalpy