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."
Use abbreviations and acronyms only when they will help your readers by making written text simpler and less cumbersome. Do not use an abbreviation or acronym that would confuse your readers, that they would not recognize quickly. When in doubt, spell it out. (An abbreviation is a shortened version of a word or phrase, like Mr. and Corp.; an acronym is an abbreviation formed from the first letter or letters of a series of words, like AIDS, Garbl, NAACP and radar.)
Always spell out terms, common names and the complete proper names of organizations, projects, programs or documents the first time you use them, and repeat the complete term or name at the beginning of sections in longer documents. Although the abbreviation or acronym is capitalized for some common or generic nouns and terms, lowercase the spelled-out form.
If an abbreviation or acronym of the term or name would not be clear on second reference, avoid using it. Instead, use a shortened version of the name or a generic word, such as the agency, the committee, the department or the company.
If using unfamiliar abbreviations and acronyms is necessary, follow the complete name with the shortened form set off between commas: The Endangered Species Act, or ESA, affects many projects. Later references could use the abbreviation, a shortened version of the name or a generic word.
Whenever possible, avoid following the name of an organization, project or program with an abbreviation or acronym in parentheses or set off by dashes: Endangered Species Act (ESA).
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Do not provide an abbreviation after spelling out a term if the abbreviation isn't used elsewhere in the document.
Omit periods in most abbreviations and acronyms unless the result would spell an unrelated word. Use only one period when a sentences ends with an abbreviation that includes periods.
Don't use the before acronyms pronounced as words instead of letter by letter: OSHA, CAD. With other abbreviations, apply the same rules for the full name and the shortened version: the ESA, the state DOT, IBM. When placing either a or an before an abbreviation or acronym, determine how it would sound when spoken; see a, an, the entry above.
To form most common plural abbreviations, add an s: ABCs, CDs, chaps., Drs., IOUs, TVs, UFOs. Sometimes, an apostrophe may go before the s: when the abbreviation has internal periods (M.A.'s, M.B.A.'s, Ph.D.'s), when the abbreviation is composed of lowercase letters (pdf's), when the abbreviation is a single letter (A's, S's) and when the abbreviation would be confusing if only the s were added (OWS's instead of OWSs). In the last example, if your readers might misinterpret an abbreviation like OWS's as showing possession, leave out the apostrophe.
Avoid using e.g., i.e.; et al.; etc.
Many abbreviations may be used in charts, tables and certain types of technical writing.
If the meaning is clear, abbreviations may be used in headlines and headings.