It was most certainly Enea Silvio Piccolomini, later pope Pius II, who came up with the cognomen 'Achilles' for Albrecht III.

Piccolomini spent about 20 years in the HRR, most notably in Vienna, where he lived off the patronage of the emperor's chancellor as a poet laureate, while Albrecht was imperial court master. They knew each other, and Piccolomini, who was 9 years older, even promoted Albrecht.

They have had some disaggreements over franconian politics but had otherwise matching views, e.g. both supported the emperor and both were in favor of a campaign against the turks when this was discussed in 1455.

Piccolomini was a humanist, the first full fledged to ever live in the HRR. Humanists attached great value to mastery of language in written and spoken form (hence the poet laureate) and also to the knowledge of classical literature.

Since many greek intellectuals fled to Italy after the fall of Constantinople and brought a large amount of greek literature with them that had previously been unknown in the west, greek became the language to be known by humanists. This will become more important later, but at this point a cognomen on classical grounds was just to be expected by a humanist.

It would later become quite fashionable for german humanist intellectuals to change their names to greek translations of their german surnames. The Neanderthal e.g. is named after a valley near Düsseldorf that was named after a certain Joachim Neander whose ancestors had changed their name from Neumann (Newman).

Also the notable humanist lutheran reformer Philip Melanchthon was originally born Philip Schwarzerdt (Black Soil) and translated his surname into greek.

John Cicero (that's a roman name btw) gained his cognomen posthumously and to my knowledge noone is really sure why. Some people say, he was a great orator and very versed in latin, others say he didn't know any latin. He had some diplomatic talent though.

To my knowledge it's also not really clear, why Joachim I was called 'Nestor'. He fought the robber barons in Brandenburg and founded in Frankfurt/Oder the first brandenburgian university, which was surely wise, but if this is all to qualify him to be called 'Nestor' we don't know. There's a cause why classical literature was so important for education in that time, people still didn't write as much as we would have liked.

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