Few dictionaries, not even the Oxford English Dictionary, give room to this word, so it is left mostly to non-lexicographers to define it, which they often do in terms such as “good writing on a trivial or base subject”. Near, but not quite right.
It’s a modern word to describe an ancient way to train young people in the art of rhetoric. They would be challenged to compose a speech praising an unpleasant idea such as poverty, ugliness, drunkenness or stupidity. So a better definition would be “rhetorical praise of things of doubtful value”. Anthony Munday published a book on the method in 1593, a translation of an Italian work, under the title The Defence of Contraries. It contained brief disquisitional examples on topics such as “ignorance is better than knowledge” and “it is better to be poor than rich”. Its preface claimed that it would be particularly useful to lawyers.
The root is Latin adoxus, paradoxical or absurd, but not from the classical language. It was first used by the Dutch scholar Erasmus around 1536, who took it from an identical ancient Greek word that meant inglorious. It was based on the root doxa, opinion or belief, which is also the basis of doxology, a formula of praise to God, and also of paradox.
The noun was first used in 1909 in The Conflict of Religions in the Early Roman Empire by Terrot Glover, though it was preceded by the adjective, adoxographical, which appeared in the American Journal of Philology in 1903. Dr Alex Leeper, the Warden of Trinity College, Melbourne, commented in Notes and Queries that year that it was an “ungainly word” and that it “will not, it is to be hoped, take root in the language.” His hope wasn’t fulfilled, though it remains rare.