I am writing an essay on the short story "Harrison Bergeron." How should I abbreviate the title if I don't want to write out the whole name? Would it simply be "Bergeron" or Harrison?"
Identifying a title by a single word from it is by no means a recent innovation, nor does it seem to be tied to the length of the original wording. Consider The Tragedy of King Lear, cited most often as King Lear, but frequently referred to simply as Lear, as in the opening paragraph of Charles Jennens, "The Tragedy of King Lear, as Lately Published, Vindicated" (1772):
As the new edition of Shakespeare's Lear was attacked in a very rude and scandalous manner, by the Critical Reviewers ; and the patron, the editor, and another person who had no concern therein (but Whom they judged to be the editor) were treated in very abusive and scurrilous terms, by this society of gentlemen, as in their title-page they are pleased to stile themselves ; it was thought proper, upon presenting another play to the public, to vindicate the said edition of Lear from the base aspersions and misrepresentations which these Drawcansirs [that is, broadswords, as opposed to rapiers] in criticism had cast upon it.
Likewise, we find The Posthumous Papers of the Pickwick Club commonly reduced to The Pickwick Papers, and from there sometimes shortened further to Pickwick, as in Joseph Miller, Reading Narrative (1998), excerpted in The Novel: An Anthology of Criticism and Theory 1900–2000 (2009):
The latter two relations, that between author and narrator, that between text and critic, are articulated with special clarity in the passage from Pickwick Papers. In Pickwick, moreover, the way all three examples exploit properties more salient in written, not spoken, language is made explicit.
And some authors regularly refer to A Streetcar Named Desire as Streetcar, as in Philip Kolin, Tennessee Williams: A Guide to Research and Performance (1998):
Cohn usefully comments on the symbology of the names in Streetcar, while Kolin explicates the mythic and gaming allusions behind Jax Beer ("Why Stanley"). Kolin also explores the network of paper signifiers in and underneath the script in Streetcar, including poetry, legal documents, and artifacts, and concludes that for Williams paper is "both script and Scripture" [citation omitted].
Of course, these short forms are helped by the fact that the works they refer to are unlikely to be misidentified by their readers—but that is surely true, too, of the short form Bergeron once you have properly introduced the complete title Harrison Bergeron to your readers. I concur with Lore Sjöberg that Bergeron, being a more memorable identifier than Harrison, would be a better choice for the short-form title.
I would caution you, however, that some readers may react unfavorably to your use of a short form of the title, as Cerberus and Edwin Ashworth do in the comments beneath your question. Also, some titles resist reduction to one word more vigorously than others do; thus for example, having shortened Moby-Dick; or, The Whale to Moby-Dick, I would strongly advise against shortening it further to either Moby or Dick.
Ultimately, your safest bet is probably to follow Cerberus's advice and refer to the title by its full name or (for variety's sake) by a descriptive term such as "the story."
To answer your actual question, if I had to abbreviate it, I'd abbreviate it Bergeron for two reasons. First, because in formal writing and journalism it's more common to refer to someone by their last name, and secondly because "Bergeron" is more unusual, and thus more memorable, than "Harrison."
(That's not a strict rule, though. I'd definitely refer to Pippi Longstocking as Pippi, because calling it Longstocking just sounds pretentious for a silly children's book.)
If the last name more common than the first, as with Pincher Martin for example, I might consider using the first name. More likely, I'd probably just refer to it by the full name. If I was tired of typing it, well, that's what search-and-replace is for.
And really, in the end, I'm not sure I'd abbreviate a two-word title at all in a formal essay, unless the first word was an article.
I had the same problem when I was writing an essay about Ursula K. Le Guin- I mentioned her novel The Left Hand of Darkness, and I needed to mention it again when I was describing The Dispossessed. My suggestion is to take out as many words as you can without changing the meaning. In my case, I shortened it to Left Hand, in your case, I would say to do Bergeron.