I am looking for the cheapest, smallest way of charging a 50-100 mAh 3.7V Li-Ion battery through micro USB. Fast-charging is not necessary, especially if it would cause an increase in price or size.
To give you some more background on my ECE education. It has focused primarily on electrical circuits and their biological applications. A really huge gap in my education is that I have almost no knowledge regarding Li-Ion batteries, let alone how to safely charge them. In fact, I have stupidly just been soldering these kinds of batteries to my board for a DIY project without putting them through any kind of regulator. Therefore, I really am recognizing my ignorance now as I begin to research charging these kinds of batteries, and would like to pose this question to those with a more traditional ECE background to make sure I don't make such a stupid, dangerous, and wasteful mistake again.
I have found a number of interesting example circuits, but most of them involve regulators designed for much higher current uses. My circuit requires about 1-5mA. The ultimate design is a heartrate monitor, so safety is also a concern. Charging time is not a concern.
How can I find an effective battery and charger design for this particular use case? Is the MCP73831 the most appropriate IC for this design? What kind of parameters and search terms should I use to find the cheapest possible product on a site such as DigiKey?
Ideally, I would prefer something that could be easily described to a foreign circuit manufacturer should I decide to go that route. That is, I don't just want a simple, overpriced charger from Sparkfun. The ideal answer would be a number of individual components (USB connector, charging circuit, battery), or a pre-made charger that is extremely cheap and needs only a battery to connect to. I want to build it myself, in a way that can be mass manufactured later on, should I be fortunate enough to go that route.