Why do we say "Every dog has its day"?

Meaning:

Sooner or later, every person has a stroke of good fortune.

Background:

It is safe to assume that this expression was in regular use during the mid 16th century as, in response to his request for a picture of her, the then Princess Elizabeth wrote to her brother King Edward VI as follows:

"…Of this also yet the proof could not be great, because the occasions have been so small, notwithstanding, as a dog hath a day, so may I perchance have time to declare it in deeds, which now I do humbly beseech your majesty, that when you shall look on my picture you will witsafe to think, that as you have but the outward shadow of the body afore you, so my inward mind wisheth that the body itself were oftener in your presence…"

This letter (and the other related correspondence between Elizabeth and Edward) was published by John Strype in Ecclesiastical Memorials in 1550, but the reference to the dog having his day appears to be from a 1549 letter.

John Heywood recorded the proverb in the 1562 edition of Proverbs and Epigrams; and Shakespeare uses it in Hamlet, 1603:

Hamlet:

Hear you, sir.
What is the reason that you use me thus?
I loved you ever. But it is no matter.
Let Hercules himself do what he may,
The cat will mew and the dog will have his day.

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