As years pass this once revengeful phrase (or as some say, idiom) has turned into a phrase of opportunity. Ie, every dog has the chance to make it. But it's original context was far more bitter.
"According to the medieval Dutch scholar Erasmus, the saying came about as a result of the death of the Greek playwright Euripides, who in 405 B.C. was mauled and killed by a pack of dogs loosed upon him by a rival. Thus the saying is usually taken to mean that even the most lowly person will at some time get revenge on his oppressor, no matter how powerful the man may be. The Greek biographer Plutarch recorded the proverb for the first time in 'Moralia' (A.D. c. 95) rendering it as 'Even a dog gets his revenge,' and Richard Taverner included the first version in English - 'A dogge hath a day' - centuries later in his 'Proverbes' or Adages' (1539).What was virtually the modern form appeared in John Ray's 'A collection of English Proverbs' (1670) as 'Every dog hath his day'." From "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).
It was also used by infamous Tony Montana. Which is actually more in touch with its original roots as a phrase of revenge.
"Even the lowliest will sometimes come to the fore, as in They may not listen to me now, but just wait, every dog has its day," (Dictionary)
But as language goes, the meaning has since evolved. "Definition: every person has a chance for success
Explanation: Used when reassuring someone that he / she will also eventually have some form of success.
Examples: Be patient, remember every dog has his day! - I told her to calm down and remember that every dog has his day." (ESL)
Also referred to as, Every Dog has his day.
Sources:
http://dictionary.reference.com/browse/every%20dog%20has%20its%20day
http://www.phrases.org.uk/bulletin_board/19/messages/1172.html
http://esl.about.com/library/glossary/bldef_414.htm