What is the origin of this phrase which seems to be trotted out every St. Patrick’s Day? What is it supposed to mean? What is the “top” of a morning, and is there a bottom of the morning? Is there something missing? For example, should it be “Fine top of the morning” which might correspond to “good morning”?
IIRC its a traditonal Irish greeting that goes:
Person 1 ‘The top of the morning to you’
Person 2 ‘And the rest of the day to you, sir.’
Meaning: The best things of the morning to you, and the best things of the rest of the day to you.
Similar to Good Day: have a good day.
From a music-hall song, c1900:
“The top o’ the mornin’ to you—
'Tis the same, sure, as ‘how do ye do,’
‘Glad to see ye’, and ‘how be ye,’
‘Have ye seen anything new?’”
Previous posters have gotten the essential meaning right, but I feel compelled to point out one thing: NOBODY in Ireland uses that phrase any more. Nobody in Ireland, or of Irish descent, has used that phrase in a loooooong time.
Comedians and cartoon-makers use that phrase as shorthand (in the same way, all comic Frenchmen wear berets, all comic Italians say “Atsa motta for you,” and all comic Englishmen wear bowlers and carry umbrellas).
So, if you’re travelling in Ireland, don’t use that phrase to impress the locals- it’ll only confirm their suspicions that you’re a typical American bonehead whose only knowledge of Ireland comes from old Barry Fitzgerald movies.
Next thing, astorian will be trying to convince us that Scotsmen don’t really say “Och, Aye, the noo” and Welshmen don’t really call each other “boyo” all the time.
Begorrah !
Say it isnt so Tom.
This leprachuan will be crying into his green guinness tonight.
FWIW The correct reply to “Top of the Morning” is not “and the rest of the day to you sir” its a muttered “ignorant yank” 
Arragh, sure isn’t it a lovely fine day we’re having today?
top of the morning to ye all and be sure to have a shilleleagh under your arm so that the road rises to meet you. If off to the crossroads to dance with some feirey haired callíns. 
Actually, I’m off to see Ulysses (finally released after 33 years banned!) with Ruadh (who, I can assure you, is a feirey haired callín) and Hibernicus (who, I can assure you, is not).
There are so many phrases in the Irish lingo for saying hello, and possibly the worst one is trotted out every Paddy’s day. Ah well, I’ll have to drown my sorrows in a Guinness!!