Typically used to describe the last encounter between two friends, lovers, or family members before a long leave of absence (or in most cases, permanent).
Note: The literal saying of goodbyes doesn't necessarily have to be drawn out. "Long" only clarifies the fact that the people involved won't see each other again for a long time (if ever).
They said their long goodbyes before Tommy headed out west.
I hate the long goodbye.
I wish I had known that was the long goodbye. There's so much more I would have liked to say.