Be systematic. This means, expect to do a lot of work...
Determine which development times and agitation schemes you will use for each developer. In doing so, you may want to standardize on a specific gamma/contrast for each film tested across all developers. After all, it won't make much sense to compare e.g. HP5+ developed to a gamma of 0.8 and to a gamma of 0.5 in terms of grain appearance, and then concluding it has something to do with the different developers used. You'd have to develop to the same gamma in both instances - or alternatively use a single developer for both contrasts to say something about grain.
In other words, you seem to be focusing on "developer" as the independent variable of your investigation, but you'll have to think about other factors that influence the end result as well ("confounders"). If you don't control for these, there will be no end to the discussion about whether you did it right (which, arguably, you wouldn't have) and the results will never quite convince.
So one of the first steps will be to establish a development process for each developer and film combination that gives the same gamma - and you'll have to do some assumptions (
and standardize on those, too) in determining gamma, since the curve shape produced by the different developers will be a little different. Reading up on sensitometry and densitometry will be required, and you will have to fashion some kind of densitometer and provide for a means to expose your film in a controlled way (this can be with a step wedge, contact printed with a known/stable light source) in order to perform sensitometry.
Another issue worth mentioning is the choice of dependent variables. What kind of differences will you be looking for? Presumably something like granularity, H/D curve shape, effective film speed, rendering of fine detail (or whatever proxy of 'acutance' you may choose) and perhaps one or two more things. Having picked those, you'll have to operationalize, which means choosing the indicators you'll use to measure/observe these variables. How are you going to observe or measure granularity in such a way that it allows for a systematic comparison?
I also concur with the comments made by others to very critically review the scanning process and perhaps even do what
@Steven Lee suggests: skip it altogether. It'll still leave you with the challenge of making the comparison visible, but in part you can solve this through measurement (e.g. curve shape, densitometry). In part, it's quite feasible in the darkroom since it's straightforward to make identical prints from similar negatives, and you won't have any black box of scanner software and firmware (auto exposure!!!) that can spoil the broth.
Putting all this in a conceptual model can help spot the (1) complexity of the task and (2) prepare for the project by identifying the various factors that can influence the end result. A partial model could look like this:
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Mind you; you can debate whether certain variables are moderator or independent variables (/antecedents), and the model is also not quite complete, since it omits for instance anything related to measurement of the end result. What it does show is the risk of the 'capital offense' of picking a developer, not minding about the rest, making some scans and then proclaim that "Developer X does Y on this film". So much happens in-between and around that (very indirect) relationship as to make any such conclusion void, really.
Alternatively, don't mind all this, just soup some randomly exposed film (street scenes or so) in whatever developers you fancy and have some fun doing it. It'll be a whole lot easier and probably a lot more fun, but please, if you choose this approach, don't draw any conclusions from it. I hope the argument above and in particular the (partial) conceptual model explains why this wouldn't work well.
If you want to proceed with the project, I'd suggest to
keep it simple and limit the project to ONE type of film and 3-4 developers, max. Standardize all the rest, and be super clear and consistent on the choice of variables and indicators and your observation/measurement methods. Consider that if you add just one more type of film to the list, the table of all possible combinations explodes since it'll be based on the cartesian product of the factors chosen. One film and four developers is four options,
provided you standardize all the rest in a sensible manner. Two films and four developers already makes eight options, and so on.
Finally, since setting up the whole shebang concerning control variables (determining development process parameters so you get comparable outcomes) and indicators, I'd suggest splitting up the project into two parts: a preparatory project in which you determine the test procedures and conditions, and a second phase where you can do the actual work. If you do phase 1 properly, phase 2 will be a straightforward taks and the results will basically just roll out of the project.
Hope this helps. Also, feel free to just go out and have fun with your camera instead. Ask yourself what the added value is of knowing that developer X gives slightly tighter grain on film Y when scanned in scanner Z. It's not going to make a better photograph. Then again, if you find this kind of systematic test fun (which I can very well imagine, and to an extent I even share that inclination), by all means go ahead.
