The phrase over the moon means very happy, delighted.

However, the earliest occurrence of the metaphor to leap over the moon has the opposite meaning; in the course of the tragicomedy The Humorous Lieutenant (circa 1619), by the English playwright John Fletcher (1579-1625), the Lieutenant is in distress:

(1811 edition)
– Gentleman 2: What think you of this sorrow?
[…]
– Gentleman 1: ’Twould make a man leap o’er the moon to see him
Act these.
– Gentleman 2: With sighs as tho’ his heart would break;
Cry like an unbreech’d boy; not eat a bit.

The phrase in its current sense perhaps originated in Ireland, since all its early instances are found in texts by Irish authors. It is first recorded as to jump over the moon in The Coquet: Or, The English Chevalier (London, 1718), a comedy by the Irish playwright and journalist Charles Molloy (1690-1767); at the very beginning of the play, Ranger, servant to Bellamy, the English Chevalier, finds his master in a prison:

’Tis he, ’tis he! I know him now: I shall jump over the Moon for Joy! Master, my dear Master, have I found you!

In a letter to the English poet Alexander Pope (1688-1744) that the Irish satirist, poet and Anglican cleric Jonathan Swift (1667-1745) wrote from Dublin on 8^th July 1733, to leap over the moon refers to youthful energy:

(1741 edition)
In one part of your letter relating to my Lord B. and your self, you agree with me entirely, about the indifference, the love of quiet, the care of health, &c. that grow upon men in years. And if you discover those inclinations in my Lord and your self, what can you expect from me, whose health is so precarious? and yet at your, or his time of life, I could have leap’d over the Moon.

The Irish author Arthur Murphy (1727-1805) used a different image in his comedy All in the Wrong (London, 1761); Sir William Bellmont is surprised that his son is unhappy to be engaged to Belinda:

– At your age I would cut a caper over the moon on such an occasion.
– Sir, I must beg to be excused—I am a little more slack-mettled, Sir, and can’t leap quite so high.

(to cut a caper: to dance in a frolicsome way – to be slack-mettled: to be lacking in energy)

The English actor and playwright David Garrick (1717-79) used yet a different form in his comedy The Irish Widow (1772). Widow Brady, who is a young woman, says:

(1794 edition)
I was oblig’d to mourn for my first husband, that I might be sure of a second; and my father kept my spirits in subjection, as the best receipt (he said) for changing a widow into a wife; but now I have my arms and legs at liberty, I must and will have my swing: Now I am out of my cage, I could dance two nights together, and a day too, like any singing bird; and I’m in such spirits, that I have got rid of my father, I could fly over the moon without wings, and back again, before dinner.

The phrase was popularised by being included in Mother Goose’s Melody: Or, Sonnets for the Cradle¹, a collection of “the most celebrated Songs and Lullabies of the old British Nurses, calculated to amuse Children and to excite them to Sleep” probably first published in 1780 or 1781:

(1791 edition)
High² diddle diddle,
The cat and the fiddle,
The cow jump’d over the moon;
The little dog laugh’d
To see such craft,
And the dish ran away with the spoon.
It must be a little dog that laugh’d, for a great dog would be ashamed to laugh at such nonsense.

¹ The name Mother Goose is from Contes de ma mère l’oye (Tales of my mother goose – 1697), a collection of fairy tales by the French author Charles Perrault (1628-1703).
² High was later altered to Hey.