The Organon by Aristotle is a set of six books. Here is an example of the use of "predicate" in Categories v (page 29)
The species is predicated of all individual examples, the genus of these and the species....For all we affirm of the predicate will also be affirmed of the subject.
In a footnote in the Prior Analytics, I. iv, the translator, Hugh Tredennick, remarks, "the predicate is naturally a more comprehensive notion than the subject".
The predicate applies or does not apply to the subject.
Regarding the sentence, "Socrates is a man", Henrik Lagerlund remarks that the use of the verb "is" originates with Boethius:
Boethius made no substantial contribution to the theory of the syllogism, though he was an important transmitter of the theory to later logicians and his works offer a clear presentation of the Aristotelian account. But that presentation differs from Aristotle's in one important respect. In Boethius, the categorical sentences are constructed using ‘is’ (’est’) and not ‘belongs’, as in Aristotle.
Aristotle. The Categories On Interpretation Prior Analytics. Loeb Classical Library. 1962. Retrieved on May 16, 2019 from Internet Archive at https://archive.org/details/L325AristotleIPriorAnalytics/page/n5
Lagerlund, Henrik, "Medieval Theories of the Syllogism", The Stanford Encyclopedia of Philosophy (Winter 2017 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/archives/win2017/entries/medieval-syllogism/.