Farewell song:
The Call - Regina Spektor

Listen to this farewell song:

This pretty song featured in the movie The Chronicles of Narnia: Prince Caspian, and is war-themed.
The meaning has been described as about those you trust in difficult times, about going away to do what is right and just.
"Pick a star on the dark horizon
And follow the light
You'll come back when it's over
No need to say goodbye."

Other songs like this one:

We'll Meet Again by Vera Lynn
Indulge in some wartime nostalgia with this classic song by Vera Lynn. 'We'll Meet Again' is a famous song of the Second World War era, and its optimistic tone a resonant comfort for soldiers and their families and sweethearts who were separated by the war.
It's a perfect song for masking a heartbreak with a smile:
"We'll meet again
Don't know where
Don't know when
But I know we'll meet again
Some sunny day..."

True Colours - Cyndi Lauper
Beautiful friendship affirmation ballad.
While not specifically a goodbye song, it is so lovely and easily dedicated to a loved one that is leaving:

"If this world makes you crazy
and you've taken all you can bear
you call me up
because you know I'll be there".

Into My Arms - Nick Cave

Written and performed by Nick Cave, this is not so much a farewell song, but rather about the return of a loved one 'into my arms', wherever they might be.

It is a deeply spiritual song despite possible claims to agnostism in the lyrics.

The above two points are expressed perfectly in the first verse:

"I don't believe in an interventionist God
But I know, darling, that you do
But if I did I would kneel down and ask Him
Not to intervene when it came to you
Not to touch a hair on your head
To leave you as you are
And if He felt He had to direct you
Then direct you into my arms"

Whether you are a parent, lover, or a very close friend, this song will certainly affect you when you hear it, if you are missing someone dear to you.