Original Meaning and Derivation of "Ever and Anon"

Solution 1:

Indeed, ever and anon goes back at least as far as Shakespeare, who used it in Henry IV, Part 1. Wikipedia says it "was almost certainly in performance by 1597" and "was entered into the Register of the Stationers Company on 25 Feb. 1598, and first printed in quarto later that year".

But how much earlier does the phrase go?

Probably not much further, as the earliest quotation in the Oxford English Dictionary is also by Shakespeare, in Love's Labour's Lost just a few years earlier:

Euer and anon they made a doubt.

The OED has the first known publishing as 1598 and Wikipedia says it is "believed to have been written in the mid-1590s for a performance at the Inns of Court before Queen Elizabeth".

Often people claim Shakespeare was the coiner of this-or-that many thousand words in the English language, but it's often the case that as a writer of many famous plays, it's more likely his use has survived the ages when someone else may have written it earlier and we've lost their text. People may have already been using them in unrecorded speech for decades. His use is often the most well-known, and dictionary compilers liked to include his quotations in dictionaries.

As a good example, I found an eight-year antedating in The Countess of Pembroke's Arcadia (1590, Book II) by Sir Philip Sidney:

But euer and anon turning her muzzell toward me, she threwe such a prospect vpon me, as might well haue giuen a surfet to any weake louers stomacke.

Solution 2:

Anon \A*non"\, adv. [OE. anoon, anon, anan, lit., in one moment)

1. Straightway; at once. [Obs.]

   The same is he that heareth the word, and anon with joy receiveth it. --Matt. xiii.

2. Soon; in a little while.

   As it shall better appear anon. --Stow.

3. At another time; then; again.

   Sometimes he trots, . . . anon he rears upright.--Shak.

4. At once; right off.

   Anon right --Chaucer.

[source]

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Ever and anon, in our case means now and then; frequently; often which can be easily confirmed if we go through the following excerpt.. [Henry IV (act 1, scene 3):]

My liege, I did deny no prisoners.
But I remember, when the fight was done,
When I was dry with rage and extreme toil,
Breathless and faint, leaning upon my sword,
Came there a certain lord, neat, and trimly dress'd,
Fresh as a bridegroom; and his chin new reap'd
Show'd like a stubble-land at harvest-home;
He was perfumed like a milliner;
And 'twixt his finger and his thumb he held
A pouncet-box, which ever and anon
He gave his nose and took't away again;
Who therewith angry, when it next came there,
Took it in snuff; and still he smiled and talk'd,
And as the soldiers bore dead bodies by,
He call'd them untaught knaves, unmannerly,
To bring a slovenly unhandsome corse
Betwixt the wind and his nobility.

Which clearly means that every now and then the fair, neat and clean person, one who is referred as to fresh as bridegroom smelled the bottle between his fingers.

Hope I have answered your query.