King Alcinous, whose counsels were inspired of heaven, was now reigning.
What does this "of" mean?
It’s an old-fashioned way of saying inspired by.
The translation you are reading was published in 1900; and its author, Samuel Butler, was an elderly man (my age, in fact). Although he was a progressive intellectually, his literary tastes were formed in the Victorian period, when it was felt that Great Literature (especially the classics, and even more especially poetry) should employ an ‘elevated’ diction—which usually meant affecting archaic words and syntax.
Inspired of God is a 16th-century usage, and is still current among conservative Christian writers. (Early Modern English is a natural idiom for them, for they cherish the 1611 'King James' translation of the Bible.) Butler, whose father was a clergyman, would certainly have been familiar with the phrase and would have regarded it as appropriately distanced from everyday language.