Could you use a low-friction pulley system to directly drive a tiny micro-DC motor as a micro-generator from a hanging weight? A typical US$3 micro-DC motor can handle about 1–2W of power and produce .49 mN·m of torque, which at a wheel radius of 5mm, is 9.8mN, the Earth weight of 10g. If your weight is, say, 10kg, you need a 1000:1 pulley system; if it is raised through 2m of distance, a simple block-and-tackle would mean you need 500 pulleys at the top, 500 below, and 1000 iterations of thread running between them, for a total of 2000 meters of fine thread. (The thread would ideally be impractically fine: 28 microns of most materials would be sufficient, for a total cross-sectional area of 2.4 mm², and much less of high-strength materials.)
To avoid this massive number of parts and threads, it’s probably better to use a multi-stage system: five sequential 4:1 reductions, for example, using gears or pulleys or whatever. A regular 40/3 cotton thread at 4 grams per denier should be able to support 2.7 kg, so maybe 4 to 16 threads running between 4 to 16 pulleys, followed by a 64:1 to 256:1 reduction gearbox or belt drive. (A 32:1 belt-drive reduction could perhaps be done in a single stage, reducing losses, with a 32-cm wheel and a 1-cm wheel; two 16:1 stages would provide 256:1.)