That is not an easy task, because it depends of too many factors. we should be able to differ what kind of aid you want to test. Food aid is very different from credit with low (or near zero) interest. So, to test it, we need to know what kind of aid is computed in the variable.

How I would try to measure it: I would build econometric models using foreign aid (or a specific kind of aid) as independent variable. As dependent variables I would test both income distribution, deaths by nutrition deficiency, GDP growth, capital formation, etc.

At World Bank Data site there are enough data for foreign aid destined to African countries, and information on development indicators.

Here is a link for it: http://databank.worldbank.org/data/reports.aspx?source=africa-development-indicators

I hope my answer helps. That is a topic I would be very interested in further discussion.

In addition to what Mr. Cosenza has outlined it guess it would be worthwhile to limit your focus based on data availability. Here are more links to comprehensive data set managed and produced by the OECD. These should help you narrow your RQ and analysis method. 

https://data.oecd.org/oda/net-oda.htm

https://www.oecd.org/statistics/datalab/oda-recipient-sector.htm

https://www.synisys.com/products/development-assistance-database/

Like mentioned above aid is not aid, there are many variations and that also has impacts on the outcome of what is achieved by the aid. When countries went through Structural Adjustment Programs they received aid to cut down on social (so-called non-productive) expenses..

Statistics are available from the World Bank (https://data.worldbank.org/indicator/DT.ODA.ALLD.CD), the OECD (https://data.oecd.org/oda/net-oda.htm) and other websites, sometimes with comments and interpretation (https://www.weforum.org/agenda/2016/08/foreign-aid-these-countries-are-the-most-generous/). To measure socioeconomic impact directly is difficult or even impossible. To draw direct relationsships or correlations between the amount of ODA and socioeconomic change can be even misleading. Whatis required are qualitative studies that look deeper into the impact of aid.

For Africa e.g. the following publications are surely helpfull:

Simone Raudino (auth.)

Development Aid and Sustainable Economic Growth in Africa: The Limits of Western and Chinese Engagements [1 ed.]

Palgrave Macmillan

2016

Zehlia Babaci-Wilhite (auth.)

Language, Development Aid and Human Rights in Education: Curriculum Policies in Africa and Asia

Palgrave Macmillan UK

2015

Engler, Yves

Canada in Africa : 300 years of aid and exploitation

Fernwood Publishing ; Vancouver

2015

Pedro Amakasu Raposo (auth.)

Japan’s Foreign Aid Policy in Africa: Evaluating the TICAD Process

Palgrave Macmillan US

2014

Danielle Resnick, Nicholas van de Walle

Wider Studies in Development Economics

Democratic Trajectories in Africa: Unravelling the Impact of Foreign Aid [1 ed.]

Oxford University Press

2013

Bronwen Everill, Josiah Kaplan (eds.)

The History and Practice of Humanitarian Intervention and Aid in Africa

Palgrave Macmillan UK

2013

Lindsay Whitfield

The Politics of Aid: African Strategies for Dealing with Donors

Oxford University Press, USA

2009

Samir Amin, Patrick Bond, Demba Moussa Dembele, Khadija Sharife, Charles Mutasa, Bernard Founou Tchuigoua, Sanusha Naidu, Tim Murithi

Aid to Africa: Redeemer or Coloniser?

Pambazuka Press

2009

Dambisa Moyo

Dead Aid: Why Aid Is Not Working and How There Is a Better Way for Africa [First American Edition]

Farrar, Straus and Giroux

2009

Jonathan Glennie

The Trouble with Aid: Why Less Could Mean More for Africa (African Arguments)

2008