Is there a word for the sadness over "What might have been"?

Solution 1:

Perhaps acceptance captures some of the sense you're seeking (see painful acceptance below).

You try, and you try, and you try, until you reach the point where you can try no more. You have slowly come to the certain realization that you will never get what you want and need, that there's nowhere to go with the relationship, that there's no future.

You have two options: continue down the road to nowhere, or call it quits. You're not going to want to call it quits unless you believe, with all your heart, that you've done everything you could to make it work: that's the only way you're going to be able to sleep at night, knowing that you did everything you could.

When at last you reach the point of calling it quits, and you make the painful decision to end it, you resign yourself to the outcome, to a new and different future. You don't want to do it, you really don't, but you have no choice. The die is cast.

That's acceptance, at least as I imagine it in the context of your question.

You could call your feeling of "what might have been" a painful acceptance. It looks both backward (painful) and forward (acceptance).

Solution 2:

Regrets, I've had a few
But then again, too few to mention
I did what I had to do and saw it through without exemption
I planned each charted course, each careful step along the byway
And more, much more than this, I did it my way

Edit

It is heavily slanted to the father-son relationship but the phrase "the cat's in the cradle", and the song to which it refers, is probably the most powerful display of this emotion I know of in American culture.

A bit more abstract is "The road not taken" which refers to Robert Frosts poem of the same name. It captures the idea but leaves the details to your imagination.

Solution 3:

As requested:

Of all the words of mice and men,
The saddest are “it might have been.”

From Kurt Vonnegut, Cat’s Cradle Chap. 123.

I had originally thought Steinbeck’s novel title Of Mice and Men evoked this couplet (which is a mashup of lines by Whittier and some by Burns), and thus effectively evoked the expression “it might have been.” But the couplet apparently was not published as such till 1963, and Steinbeck’s novel came out in 1937. Alas . . . do I actually have to say it?

Solution 4:

“Our unfinished symphony” lacks the notion of absolute finality that you seek (after all, until one of the parties to the relationship/composers of the symphony dies, they could theoretically get back together and finish their symphony), ...

... but to the extent that one really wanted to, they could add to or paraphrase it to remove all doubt, perhaps as follows:

Grieve not for what might have been. Rather please try to think of it fondly as “our forever unfinished symphony.”;

Grieve not for what might have been. Rather please try to think of it fondly as “our never-/ne'er-to-be-finished symphony”;

Grieve not for what might have been. Rather please try to think of it fondly as "the never-/ne'er-to-be-composed notes/measures/lines/finale of our unfinished symphony.

(‘Unfinished Symphony’ by Richard Schweitzer from musicxray[dot]com)

(example of “never-to-be-composed” used in a sentence with “what might have happened”; from ‘Opera in History: From Monteverdi to Cage’ by Herbert Lindenberger, via Google Books)

(example of “unfinished symphony used in a sentence with “what might have been”; from Suzanne Strutman’s Introduction to Thomas Wolfe’s unfinished novel, ‘The Good Child's River’; via Google Books)

With the added tag "poetry" in mind:

Along the general musical lines of the above variations on the theme of “unfinished symphony” (but admittedly, more comparable to @CandiedOrange’s "The road not taken"), you could consider the idea of “the/our unsung song” or preferably, as inverted and used in the title of a book by Monique Ritter: “The Song Unsung.”

The provided link is to the website of the novel’s publisher, The Strategic Publishing Group, where a synopsis uses the notion of “what might have happened on another path” (in direct reference to the title of Mr. Frost’s poem) in its first paragraph, as well as both CandiedOrange’s “Regrets” and "The road not taken" together in paragraph 3.
(I fear that this edit is reading more and more like a reason for giving favorable consideration to CandiedOrange’s good answer than for doing so to my answer!)

Anyway, in response to your addition of the tag “poetry” (not “famous poetry,” mind you), in addition to the book mentioned above, I’ve also found a nice poem with the same title:

“The Song Unsung” by Abdul Wahab

So many people have sung so many songs
On and from earth to heaven
but they have not sung me
I am the song
Unsung

Though I have the best beat
I move with a great rhythm
Yet I have not been put on music

The reason I know not
Nor I want to know
As it could be anything
But I am sure that it is not their ability
Neither the quality which I carry

If any one sings me
Instantly he or she will be
On the cloud nine
As I am the song
None has ever touched me
Purely unsung

Oh singers of the world
If you sing me
I shall make your fame spread so wide
That your name can not hold
As I am the song unique

And totally unsung.

Abdul Wahab

(from PoemHunter[dot]com)

Although this poem, as most poems, is subject to different interpretations, as I interpret it, the poet is saying that the song that we leave unsung in our lives would be/have been far better than the one we chose to sing ... the one we are now singing (at least in our own minds, in retrospect from our own present lives).

If my pessimistic interpretation is correct (and I hope it's not, even if that would lessen the poem's relevance to your question), the poet’s message is sad and very sobering.

Solution 5:

I would suggest Hauntology, specifically as defined by cultural theorist Mark Fisher, who "used the term to describe [contemporary] art preoccupied with this temporal disjunction and defined by a "nostalgia for lost futures"."

Quote is from wiki, Concept is from Fisher's book Ghosts of my Life.