There is a strong case to be made that research into family planning should be a priority for GiveWell and other similar organizations (Open Philanthropy, Rethink Priorities, Founders Pledge etc.).

The Copenhagen Consensus Center has identified universal access to contraception/family planning as one of the most cost-effective ways to improve global welfare, in 2 separate analyses, with estimated extremely large returns for each dollar spent. In fact, excluding projects that can mostly only be implemented by governments (free trade and R&D), the Copenhagen Consensus estimated contraception/family planning to be the most cost-effective interventions possible. The first estimation, from the Post-2015 Consensus Project, estimated 120$ returns for every $ spent on universal access to contraception. The second, from Best Buys for Africa, estimates 94$ returns for every $ spent on family planning.

Apparently, Charity Entrepreneurship also estimated the return on investment of postpartum family planning, at 105$ for every $. They wrote, in a report on postpartum family planning: “Through our research, we found that this idea is among the strongest from the perspective of evidence base, cost-effectiveness, and execution difficulty”.

The Post-2015 Consensus Project also supplies us with an estimation of the relative effectiveness of family planning to direct money transfers. It estimates that 5$ are generated for every $ in money transfers to end extreme poverty. Thus, on a very rough estimate, family planning is between 18.8 to 24 times as effective as programs such as GiveDirectly’s cash transfers. This is well within the margin of investment for GiveWell, and justifies further research into family planning. If we consider the prevention of miscarriages and other health issues, it may even be higher.  

I know that these cost-benefit estimates have many limitations, and that they also depend on effective charities in that area of work, but there are 3 main considerations that show this area is still very worth investigating. First, the external validity of the aforementioned estimates is probably reasonable. Past priorities of the Copenhagen Consensus align quite well with the charities GiveWell has historically supported (I can elaborate further if needed), and Charity Entrepreneurship works within a similar framework to GiveWell. Second, the only current research article on contraception I’ve found in GiveWell, on Sayana Press, is very neglected - It does not mention the demographic dividend and some other benefits of contraception, and hasn’t been updated since 2017. Third, If family planning will be found by GiveWell to be an effective cause, and GiveWell will be vocal in support of family planning, that could greatly incentivize existing charities to extend further resources to the topic, and inspire people to donate more to that cause. Alternatively, if GiveWell will conclude that family planning may be an effective cause, but more research is needed, that could incentivize more research. Even if no current charities in family planning are good enough, GiveWell can accelerate progress in the field.

For what it’s worth, Melinda Gates also said that contraception is the “greatest anti-poverty tool in the world”. While the Gates Foundation does have massive support for family planning, that does not mean they made a significant effort to identify the most effective charities working in the field, like GiveWell would. Some forms of family planning are likely much less effective than others, as seen in Charity Entrepreneurship’s reports (potentially more than an order of magnitude).

Mechanisms of benefit: 

I will only briefly describe these as I absolutely trust interested people to perform their own research. Bjørn Lomborg, the head of the CCC, wrote that about 40 dollars of expected benefit from contraception access will come improved health associated with less births, while the additional 80 dollars will come from the “demographic dividend”, caused by parents and the local government having less people to take care of, and more revenue (because of increased work time) [source]. 

Additional benefits may come from climate change mitigation. Although today poor countries aren’t responsible for much GHG emission, these numbers are expected to increase as they get richer. Currently, Charity Entrepreneurship estimated that through family planning, 3 tonnes of CO2 can be mitigated for 1$. This is in line with the Clean Air Task Force, as estimated by Founders Pledge, and may be even better than them. 

 From a quick search, some more research and support for family planning can be found here, here, here.

In conclusion, family planning may be a very cost-effective way to improve the lives of women and children in the developing world. GiveWell and similar organizations should research family planning so that they can provide more information about this very promising intervention.

 *Other organizations that could benefit from research on family planning are Open Philanthropy, Giving What We Can, Happier Lives Institute, Rethink Priorities, Founders Pledge and Evidence Action (The Life You Can Save and Charity Entrepreneurship already have pages on contraception).
 

Some additional considerations: 

One other source - “Family Planning and the Burden of Unintended Pregnancies”, estimates the health benefit-cost ratio much lower, at 2-9$/$, and on average 8$/$. I don’t think it estimated the benefits from the demographic dividend.

Obviously there's a need for a choice and for some reproduction on the population level, but the initial implementation will not be able to hurt population reproduction in a significant way. Might need to make sure that women aren't forced to take contraception by others, although I don’t think that’s usually a problem. 

It’s worth checking what are GiveWell current donators' opinions on family planning, and the opinions of potential donators.

Family planning is relatively relatable to people in developed countries, especially with the recent turnover of Roe v. Wade. It might get more support than most other charities for that reason, which may even increase donations to GiveWell in general.

If Charity Entrepreneurship is already doing research on the matter, why should GiveWell? CE’s work seems good, but GiveWell is much more well known - searched about 12 times as much last year, according to Google Trends. The Life You Can Save is searched more than GiveWell, but the page on contraception is hidden in the site (it’s in the site's blog), and the charities recommended don’t have much research behind them. In addition, GiveWell is probably the most well regarded research site on effective charities in global health.