A to Ros. I agree that middle (even other given) names can be an important genealogical clue. In one of my lines, for example, it helped trace the maiden name of a spouse, b. 1829, because her father's middle name was somewhat unusual and, once we found him and identified that name, puzzle pieces fell together because the name had been repeated multiple times as given and middle names over the course of generations all the way to the present.
And then there's the great complication to certain Scandinavian and British Isles one-name studies: patronymics or matronymics.
However, attempting to use anything but a surname and its known variants in a one-name study as a top-level classification rather flies against the very definition of a "one-name study." The good news, though, is that you can start with the dominant (or etymologically identified) surname in the current era, and then have all sorts of flexibility in how you organize and work with the people, past and present, who either bear or are most closely associated with that name.
Ros's example of subcategorization would work, but like her I'm not certain that simply having names and profile links showing up on multiple Category pages would really help with the intent. Listings by themselves don't really describe the relationships and offer any insight into how they formed or were determined.
The Freespace page idea, though, could give you all the latitude you need, and if there are only two or three middle names that are important to name study, you might opt to use only one Space page to describe all of them and link to the appropriate profiles. That way the associations can be made clear, diagrams could be used, photos can be included, and comments could be added under specific profile links, all to help explain the relationships and the relevance.
It would take a bit more work to maintain, but probably not exorbitantly so. When you search for a specific surname, the listings show middle names and maiden names if they're included on the profile. There's also a search box on the page that's labeled "First Name." It will actually look for first or middle names, and the results are constrained to the same surname you originally searched for. As an example, if you go to the Armstrong surname listing, you'll see that 11,893 profiles are found. But if you have a particular interest in any Armstrongs whose names also include Wainwright, you can enter "Wainwright" in that "First Name" search box, and voila!, you find three Armstrongs who have the middle name Wainwright.
If they match up to your one-name research, you can add and comment on them on the Freespace page and then, as Ros noted, add a link to their individual profiles so that folks know some nifty additional information is awaiting them elsewhere.