One song, no. But here are three, that i know of.

BIRTHDAY OF ST. PATRICK
Lyrics: Leo McCaffrey

On the eighth day of March it was, some people say
That Saint Patrick at midnight, he first saw the day
While others declare 'twas the ninth he was born
But 'twas all a mistake between midnight and morn

Mistakes will occur in the hurry and shock
While some blamed the baby and some blamed the clock
Till with all their cross questions, sure no one could know
If the clock was too fast or the child was too slow

Now the first faction fight in old Ireland, they say
It was all on account of Saint Patrick's birthday
Some fought for the eighth, for the ninth more would die
And who wouldn't see right, sure they blackened his eye

Till at last both the factions so positive grew
They each kept a birthday, so Pat then had two
Till Father Mulcahy, who showed them their sins
Said no one could have two birthdays, but a twins

Says he, "Boys, don't be fighting for eight or for nine
Don't be always dividing, and sometimes combined
Combine eight with nine, seventeen is the mark
So let that be his birthday". "Amen," says the clerk

“If he wasn't a twin, sure our history will show
That at least he's worth any two saints that we know.”
Then we all got blind drunk, which completed our bliss
And we keep up the practice from that day to this

ST. PATRICK'S DAY
Lyrics JOHN MAYER

Here comes the cold
Break out the winter clothes
And find a love to call your own
You - enter you
Your cheeks a shade of pink
And the rest of you in powder blue

Who knows what will be
But I'll make you this guarantee

No way November will see our goodbye
When it comes to December it's obvious why
No one wants to be alone at Christmas time

In the dark, on the phone
You tell me the names of your brothers
And your favorite colors
I'm learning you
And when it snows again
We'll take a walk outside
And search the sky
Like children do

No way November will see our goodbye
When it comes to December it's obvious why
No one wants to be alone at Christmas time
And come January we're frozen inside
Making new resolutions a hundred times
February, won't you be my valentine?

And we'll both be safe 'til St. Patrrick's Day

We should take a ride tonight around the town
and look around at all the beautiful houses
something in the way that blue lights on a black night
can make you feel more
everybody, it seems to me, just wants to be
just like you and me

No one wants to be alone at Christmas time
Come January we're frozen inside
Making new resolutions a hundred times
February, won't you be my valentine?

And if our always is all that we gave
And we someday take that away
I'll be alright if it was just 'til St. Patrick's Day

ON ST PATRICK’S DAY
(to the tune of "Skip to My Lou")
Author unknown

Let's wear green and dance a jig,
Let's wear green and dance a jig,
Let's wear green and dance a jig,
On St. Patrick's Day.

Let's find a clover with four leaves,
Let's find a clover with four leaves,
Let's find a clover with four leaves,
On St. Patrick's Day.

Let's look for leprechauns on their way,
Let's look for leprechauns on their way,
Let's look for leprechauns on their way,
On St. Patrick's Day.

Let's search for gold at the rainbow's end,
Let's search for gold at the rainbow's end,
Let's search for gold at the rainbow's end,
On St. Patrick's Day.

Hope this helps a little.