The most common interpretation of this expression is that any given person's moment of glory is inevitable.
This proverb was used in Shakespeare's "Hamlet" (Act 5, Scene 1)
Let Hercules himself do what he may,
The cat will mew and dog will have his day.
Everyone likes to note this, but that's not where it originates. It was used in pre-Shakespearean days.
In "Wise Words and Wives' Tales: The Origins, Meanings and Time-Honored Wisdom of Proverbs and Folk Sayings Olde and New" by Stuart Flexner and Doris Flexner (Avon Books, New York, 1993), the history of this proverb is traced to the medieval Dutch scholar Erasmus.
Erasmus said that in 405 B.C. Euripides, a Greek playwright, was mauled and killed by a pack of dogs loosed upon him by a rival. So the saying is usually taken as "even the most lowly person will at some time get revenge on his oppressor, no matter how powerful the man may be."
Plutarch, a Greek biographer, recorded the proverb for the first time in 'Moralia' (A.D. 95): 'Even a dog gets his revenge'.
Richard Taverner includes the phrase in 'Proverbes' or Adages' (1539) as the first English version: 'A dogge hath a day'.
In John Ray's 'A collection of English Proverbs' (1670) it was further modified almost to what it is now: 'Every dog hath his day'.
Personally, without knowing how others define it and without hearing it in context, I always thought that this proverb means "every dog has a finite life span, and will eventually be dead." Everything above contradicts that, so I guess there is no shread of truth in what I thought.