"Corollary" is similar to the word "bonus": a little extra (i.e. an extra proposition coming from a demonstration).
The term Euclid uses is πόρισμα "porism," which Liddell-Scott-Jones cite as akin to πορίζω in the sense of "to find (money)." For instance, after I.15:
ἐκ δὴ τούτου φανερὸν ὅτι, ἐὰν δύο εὐθεῖαι τέμνωσιν ἀλλήλας, τὰς πρὸς τῇ τομῇ γωνίας τέτρασιν ὀρθαῖς ἴσας ποιήσουσιν.
The earliest usage in English cited by the OED is Chaucer's translation of Boethius, De consolatione philosophiae, Bk. III, who proposes to call a porism a corollary. Boethius is making a theological argument, not a mathematical one, so he is using corollary as analogous to the mathematical term porism. Chaucer adds "or a mede of coroune", that is, "a reward of crown," as an explanation† of corollary:
ryȝt as þise geometriens whan þei han shewed her proposiciouns ben wont to bryngen in þinges þat þei clepen porismes or declaraciouns of forseide þinges. ryȝt so wil I ȝeue þe here as a corolarie or a mede of coroune....
Super haec, iniquit, veluti geometrae solent demonstratis propositis aliquid inferre, quae πορίσματα ispi vocant, ita ego quoque tibi veluti corollarium dabo.
LSJ also cites Pappus as using porism in the sense "a kind of proposition intermediate between a theorem and a problem."
†Perhaps "meed of crown" was a phrase in the Middle Ages. I'm not familiar with it. Meed can mean reward or bribe or share (of money, honor, etc.). Possibly "crown" refers to king, lord, and by extension God in this case.