Note: I misunderstood the question, and responded to mediant keys instead of mediant pitches. But I'll leave this up in case it's helpful.
David Kopp, in his book Chromatic Transformations in Nineteenth-Century Music, offers a naming system.
He separates these mediants into two broad families of "lower" and "upper" (L and U) mediants.
Within these two families, there are then three more specific types: "flat," "relative," and "sharp" (F, R, and S) mediants. AImportant to remember is that the relative mediants share two common tones, while the other mediants share just one.
So in C major, the LFM ("lower flat mediant") is A♭ major, the LRM ("lower relative mediant") is A minor, and the LSM ("lower sharp mediant") is A major. Meanwhile, the UFM is E♭ major, the URM is E minor, and the USM is E major.
In C minor, the LFM is A♭ minor, the LRM is A♭ major, and the LSM is A minor; the UFM is E♭ minor, the URM is E♭ major, and the USM is E minor.
There are more specifics regarding distinctions between chromatic and disjunct mediants, but this is the gist of his system.
I should note that this terminology is not that widely used. While most "academic" music theorists should be familiar with this terminology, I haven't really seen it used outside of academia. But that doesn't mean you shouldn't use it!
And Tim is right on with the distinction between subtonic and leading tone, so I didn't discuss it here.