When currently writing a proposal, I find myself often using 'research approach' and 'approach to research', but I cannot come up with a suitable alternative to avoid tedious repetitions. I'm trying to describe how research is conducted in different fields quite broadly (so including theoretical background, empirical methods, "institutional memory" etc), so 'research methods' is too specific, but 'research philosophy' feels to vague.
I'd vote for
research methods or
research methodology here but if that does not fit your needs for any reason, you could use these terms which offer a broader coverage:
The phrase research approach is much less specific than research method.
To me, it gives the gist of the method, as well as including elements that don't directly fit into the method proper. It's a high-level term and IMHO overused and jargony.
For example, "Our research approach to studying X requires that we travel and hire extra staff". Here, you would not use "research method" because you'd never actually include these two facts in a methods section of your publication.
I think approach to research is fine when comparing how things are done in different fields. For example, physics takes a more quantitative approach to research than...
I would avoid research system and research paradigm. And definitely don't use research framework. We don't need more frameworks.