Are there 200 guys here who need a .405 WCF or a .257 Roberts? Mebbeso, if we notified several other of the larger gun enthusiast websites. That being written, I don't know how far out you could get in the choices.
You would likely have good luck if it is for a cartridge that requires a minimal amount of fuss on the part of H&R/Remington. For example, I kind of doubt they would be interested in turning out a .32-20 or a .405, as cool as those woud be. Things that could be far more likely are .257 or .250-3000, 'cause they already have .25-06 barrel blanks. Or longer .30-30 barrels, like 24" or 26" (I'm FIRST for that!). A .30-40 Krag is easy, etc, etc.
Wildcats are absolutely out, so no AI chambers, also no odd European things like 9.3mm.
Lets start easy. How about Tim chooses a half dozen obvious choices, and we agree to choose one from those. No thread drift, no chest beating. Then if we can decide on a single choice to begin with (doubtful), we poll how many people will actually buy the darn things. We take that to Buds or who ever and see how we can sweet talk them into ordering 200 rifles they may get stuck with. Or not.
If it goes well, perhaps it could be a yearly program through the distributor, similar to the DU guns.
My choice is a Target Classic in .30-30, 26" minimum barrel or a .338 Marlin Express, 24" barrel, nickel/SS, laminated
Whadda think Tim?