I think given the size of the cell it's not going to provide a bunch of power, so really you want one that provides enough voltage to charge the battery directly, rather than with a step-up converter.
A few things come to mind:
- Use an off the shelf solar power bank like this one - the 5v output may draw too much power to use consistently so you may need to crack it open and connect to the LiPo directly.
- As others have said, charging a LiPo from a solar cell is a bit iffy. However if you're using a LiPo with protection circuitry in it then you might be ok, especially if the cell is small so doesn't provide much power. I actually have some devices here where I do exactly that, but I couldn't recommend it for a product.
- You could use a Puck.js with a LiPo CR2032 cell, and a trace cut on the board and diode added as I mentioned in your other post (because otherwise the voltage is too high). There's no cell protection circuitry on those so I'd recommend you use a 3.9v zener diode (the closest zener voltage below 4.2v) to stop overcharging. It'll work and be safe, but I'm not convinced about the cell lifetime.
- NiMH cells are probably a good bet for this kind of thing as mentioned previously.
- You could use a supercapacitor, which should have a very long life. You just need to be sure to choose one that doesn't have a high self-discharge rate. Trying to get one that'll fit in a Puck.js form factor could be difficult.
However with all the battery based solutions (so not supercapacitor) the batteries have a lifetime - they'll only work for a certain number of charge/discharge cycles or amount of time.
If you just stuck a non-rechargeable Lithium AA cell in the box then that'd provide over 2500mAh and a 10 year life, and with simple advertising would probably outlast most of the other solutions, without needing daylight.