What does Hippocrates/Ἱπποκράτης literally mean? Horsepower?(
) Or horse reining?
Something like Horse-strong. There were lots of names with ἱππ- as first or second element, and κρατ- likewise (Hippolytus, Xanthippe, Socrates, Isocrates, Cratippus, etc. etc.).
In Aristophanes’ Frogs line 429 we also have someone called either Ἱππόκινος or Ἱππόβινος; my Loeb has the name in the first version in the text but apparently it’s an emendation, the manuscripts having the second form (if I understand the apparatus correctly). Whichever version we prefer, the meaning is the same - Horsefucker. Apparently it’s a joke played on someone called Ἱππόνικος (Horse-victor?).
I’m sorry I brought this up.
And in Aristophanes’ The Clouds, the horse-racing crazy son of a rural farmer and a sophisticated city woman is named “Pheidippides” which was a compromise between an aristocratic name containing “hippos” which the mother wanted and
farmer’s father’s name “Pheidon” (thrifty or miserly). So Pheidippides is a miserly aristocratic name.
To me, Ἱπποκράτης seems awfully similar to common Homeric epithet, ἱππόδαμος. And the common Sanskrit mythological king name Ashvapati is “Horse lord”.
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From the Greek anthology:
ΑΔΕΣΠΟΤΟΝ
Ἰητὴρ μερόπων, Ἱππόκρατες, ἀλλὰ καὶ ἵππων,
Σώσανδρε, κρυφίης ἴστορ’ ἀκεστορίης,
ἢ τέχνην μεταμείψατ’ ἢ οὔνομα· μηδὲ καλείσθω
ἅτερος ἐκ τέχνης, ἧς ἕτερος κρατέει.
The joke is that their names are switched, Sosandros, healer of men, is a horse doctor, while Hippocrates is “master” of the art of healing men.
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Later Greeks may have associated the word with the centaur? Here is Theodore Prodromos, using the word as a plural noun, which he places next to σάτυροι, μοῦσαι, νηρηίδες:
Ἔρωτες πλήττουσιν χορδάς, σιγῇ κιθαρωδοῦσιν,
δοκοῦσι παίζειν σάτυροι, σκιρτῶσιν ἱπποκράται,
αἱ μοῦσαι συγχορεύουσι, πηδῶσι νηρηίδες,
ὄρνιθες ὑπερίπτανται, κυνηγετοῦσιν ἄλλοι
τῆς Ἰνδικῆς τὰ χρύσεα πτηνὰ συναναπτάντα.
Also compare ἱπποκράτωρ.
Don’t be sorry to have brought this up. It’s absolutely hilarious! 
Cheers.
Again on the Frogs passage regretfully referenced by Paul D.: it’s a culminating jest aimed at the fabulously wealthy Kallias son of Hipponikos, the illustrious father’s name here perverted to Hippobinos, “Horse-fucker” as Paul says. Here he’s said to fight in a sea-battle wearing “a lion-skin of cunt” (κύσθου λεοντῆ), a trophy of his womanizing conquests akin to the lion-skin sported by Herakles.
And the Byzantine poem quoted by Joel is evidently an ecphrasis, a description of scenes on a painting (whether real or imaginary). That explains why the Erotes are paradoxically described as playing the lyre in silence (σιγῇ), and the satyrs only seem to be at play (δοκοῦσι παίζειν). The set-up of Longus’ delightful Daphnis and Chloe offered a model.
ἱππο- in ἱππόβινος could be taken as object or subject. If subject, it would be, instead of horse-fucker, something like ‘fucks like a horse’. We say “hung like a horse.” Various scholia suggest ὁ μέγας πόρνος as a gloss, which rather than a suggestion of bestiality, would agree with this. LSJ gives ἱππόπορνος as an equivalent, which is again apparently to be read as subject.
And bringing it back around to centaurs, Diogenes comedically describes a ἱππόπορνος as a type of centaur: ἕτερον δέ τινα ἐπὶ ἵππου ἰδὼν παραπλησίως ἔχοντα καὶ μεμυρισμένον καὶ τούτοις ἀκολούθως ἠμφιεσμένον, πρότερον μὲν ἔφησε ζητεῖν τί ἐστιν ὁ ἱππόπορνος, νῦν δ’ εὑρηκέναι.
And yes, the title that I didn’t quote is Εἰς τὴν σεβαστοκρατόρισσαν, ἐπὶ τῇ σκηνῇ αὐτῆς ζῶα διάφορα ἐχούσῃ ἐντετυπωμένα. Here’s the first hit on Google, Michael Jeffreys’ article describing it as an “ekphrasis” on sebastokratorissa Eirene’s tent.
I notice, looking up Jeffreys’ 1994 edition of the poem, that he agrees with me on ἱπποκράται. His note is almost precisely what I suggested above: “Probably centaurs (cf. LSJ, s.v. Ἱπποκράτωρ)”
A point that the various scholia repeat about the ἱππο- prefix in the Ἱπποβίνου context is that ἵππος γὰρ ἐπὶ μεγάλου λαμβάνεται. They added “horse-” to mean something like “one who bigly (like a horse) ” rather than “ a horse”.