First, for Schneidewin. In his 1838 Delectus poesis Graecorum elegiacae, iambicae, melicae (pp.16-17) he declines to include this emmendation attributed to him. So, either he saved this speculation for a later edition or published it in some paper I haven't been able to find.
Next, of course, comes Bergk. In his 1866 Teubner Poetae lyrici Graeci (p. 412) quite a lot gets said —
V.11 ἑτέρων VL, ἑτερέων BP, conieci σφετέρων vel προτέρων, Schneidewin ἱερῶν vel πτερινῶν vel ἐπεβήσεθ’ ἑῶν, Ahrens στερεῶν vel ἐπεβήσετ’ ἄρ’ ὧν.
Well. Except Ahrens' ludicrous στερεῶν, these emmendations are strikingly banal. But what problem are they trying to fix?
My email correspondent and his colleague were concerned about the sense of ἑτέρων. Helios, sleeping the night away, has been born along by the waves in a golden, winged bed made by Hephaestus. When he arrives in the land of the Ethiopians, "where his swift chariot and horses stay," he gets on his other (ἑτέρων) chariot. The question is, what other chariot? Where's the other one? Some of the emmendations seem to be inspired by this same discomfort — σφετέρων, ἑῶν, ἱερῶν, κτλ. One of Bergk's emmendations, προτέρον, seems more concerned with the hiatus, since "his earlier chariot" doesn't remove that extra chariot from the picture. Hiatus for a long vowel in princeps position is sanctioned by both Homer and other elegiasts, but without more information it's hard for me to know for sure what motivated Bergk here.
When I first read this line, I had a passage from Hesiod's Theogony in my mind, 746-757, which describes Day and Night passing each other on the threshold to the same house each day. While Hesiod mentions no vehicle for them, chariots taking celstial divinities across the sky is a common idea across Indo-Eurostan. A little digging shows that Dawn herself, mentioned in line 3 of Mimnermus' poem, is given her own chariot in the Odyssey (23.243-246), and several times in Vedic and Avestan literature (M.L. West, Indo-European Poetry and Myth, p.223.). We don't know enough about Mimnermus' model of the celestial mechanics, but it's certainly possible that the other chariot refers not to another of Helios', but someone else's.
Finally, the word I've been translating chariot, ὀχέων (a funky heteroclite in Homer, ὁ ὄχος, τὰ ὄχεα, but normal 2nd. declension plural in other authors), has a wider range of meaning. While chariot is certainly the common sense, it can mean anything which holds or carries something. In Odyssey 5.404 it describes harbors, λιμένες νηῶν ὄχοι. And it can even describe a ship, ὄχος ταχυήρης "a swift-oared vehicle." The related word ὄχημα covers the same range, from chariot to ship to vehicle. Right now I'm inclined to see ἑτέρω ὀχέων being contrasted not to some other horse-drawn conveyance, but to Helios' splendid sea-faring bed. In any case, I see no good reason to meddle with the paradosis.