I think it reasonable to conclude that indeed Styx could swear by her own name. The basic criterion making one eligible to swear this oath is for the swearer to be one of the gods, which she definitely is.
Hesiod's Theogony suggests that she would also be required to participate in a libation ceremony in which she pours her own water out as an offering, apparently to herself but also to fellow inhabitants of the Underworld who are older than herself. Homer's Iliad suggests that these would be her uncle Kronos [Cronus] and the other Titans imprisoned in Tartaros [Tartarus], who are worshipped by the Olympians similarly to how certain humans venerate their ancestors.
Hesiod never explicitly says that the swearing god must drink Styx water during the libation but it is typical of such an offering (as it is of other types of sacrifice as well), especially in Ancient Greece, for the devotee to partake of a portion of what is offered. This—drinking her own water—would be, to me, the only particularly weird thing that Styx would have to do in order to swear the dread oath containing her own name (assuming that it's actually so mandatory to begin with).
Examples of Styx oath formulae sworn by certain deities in mythography, suggest, however, that the Olympians held the names of the most ancient members of their kind, the primaeval divinities who formed portions of the structure of the universe, to be just as sacred as that of the Underworld River of Abomination.
In sources as old as the Iliad, the Odyssey and the Homeric Hymns, we encounter things like Hera swearing to Zeus, in the same oath, not only by Styx, "the greatest, most formidable oath among the blessed gods," but also by Gaia (Earth), Ouranos [Uranus] (Heaven/ Sky) and by Zeus' own head! (See Iliad 15.36.)
In the Homeric Hymn to Delian Apollo 84, the goddess Leto, while making a promise to the island of Delos, uses Hera's Iliad formula verbatim:
Now let Gaia be my witness, and the wide Ouranos above us, and the
down-flowing waters of Styx...
And it isn't only the gods who actually lived on Mt Olympos [Olympus] who used this oath. In Odyssey 5.182, the sea-nymph Kalypso [Calypso], makes an oath to Odysseus in what gives us yet another instance of the formula we've seen being used by Hera and Leto in other Homeric literature: by Earth, Heaven, and the waters of Hate.
In much later mythography, Nonnus' Dionysiaca—which, where it can, likes to allude to (or imitate) the Iliad—has Aphrodite causing Poseidon and Dionysos [Dionysus] to swear "by Kronides [Zeus] and Gaia, by Aither [Aether] and the floods of Styx" (42.526-528).
The formula has essentially been duplicated, with the name of Kronides standing in for Zeus's head (from Iliad 15), and Aither replacing Ouranos. Aither, the clear, pure air above the clouds, which only the gods could breathe safely, is a primordial deity who, though associated with the Sky, is older than Ouranos (or thus he seems to appear to be in the Theogony). Meanwhile in Homeric poetry Gaia's name always comes first, but here Nonnus has given priority to Zeus.
Logistics of Importance
Conceivably, if Styx needed to swear a solemn oath, she could simply invoke Earth, Sky (or Aether) and Zeus, leaving her own name out. Nevertheless it is not absurd for a deity to invoke her/his own name in such a ceremony. Or at least it is not completely unheard of; and the best example for this that I can think of comes from the Bible, in which a similar principle as that expressed in the aforementioned Greek texts applies.
The idea is to call upon the highest, most revered or most powerful witness to acknowledge one's oath, as an indication that one unequivocally means to keep his word. In Hebrews 6.13-14, God invokes himself to witness the promise He makes to Abraham: "since He could swear by no one greater, He swore by Himself" (which in turn is a reference to Genesis 22.16 & Exodus 32.13).