What are some formal methods in measuring socio-economic impact or broader impact of research topic or individual research papers?
There are interdisciplinary researchers who study such questions.  It's labeled with various names: "Sociology of Science", "Science, Technology, and Society" (STS), and similar. You should look at journals in those fields.  For example, Science and Public Policy has these articles:


"Opportunities for impact: Statistical analysis of the National Science Foundation’s broader impacts criterion"
"The state of the art in societal impact assessment for security research"
"Introduction: Future pathways for science policy and research assessment: Metrics vs peer review, quality vs impact"