The answer, sadly, is nothing.
Not even if I inscribed it in blood with a gold-tipped fountain pen across the front of an original Andy Warhol Marilyn print.
Unlike a job application, a cover letter for a story submission is not a showcase to demonstrate that you're not just what's in the resume. A resume is a document in a standard format designed to impart information quickly and efficiently; the cover letter is the opportunity to stand out as being interesting and exceptional among a sea of nearly identical resumes.
In the case of a story submission, the cover letter is the means of communicating certain specific information that the editor wants to see, and to give enough additional information that if a story is interesting, the editor can find out more. It is the story that has to stand out as interesting and exceptional. The roles are reversed.
I ended up settling for a fairly generic template, polite and to the point. Early on, while I was still getting my brain around what worked and what didn't in storytelling, I included a sentence that hinted that I was not adverse to receiving feedback. A number of editors who have publicly stated that they don't give feedback were kind enough to to make an exception to their policies, and it really was valuable.
Now, editing the Journal of Unlikely Entomology, I can restate the earlier proposition with a bit more authority. There is very little you can do in a cover letter to help your story get accepted. The story will stand (or fall) on its own, based on many factors, including writing quality, proper fit for the market (and that's a post all of its own, right there), strength of the story, and the personal taste of the editors.
There is, however, quite a bit you can do to shoot yourself in the foot.
It starts with the salutation. Our particular zine has two editors, and we make story selections by consensus. An author who begins a cover letter with, "Dear Mr. Mojzes," isn't winning points on Alison's scorecard, and calling her "Mr. Wise" is similarly ineffective. (I'm not sure why, but I've never been left off in that way. Perhaps it is because my name came first in the alphabetically arranged list of editors, or perhaps it is because I'm male and thus inherently (ahem) more important.) It may seem trivial, but part of your job as a writer is to get this right.
I've found the easiest way not to fuck up like this is to use a generic "Greetings," as my salutation, unless I personally know the editor it is going to. Others use, "Dear Editors" or somesuch. All those things work. Leaving out a salutation, however trivial this may seem, gives an impression of rudeness, so it's probably wise to avoid that.
Don't screw up the name of the market. I've done that - misspelled the name of a zine I was submitting to (I actually never heard back from them.). We've also seen cover letters that were clearly recycled, and the author forgot to change the name. Not something that we take offense to, and it does give us a secret glimpse into who you've submitted to before, but it hints to us that you perhaps do not give as much attention to detail as might be desirable.
Also, I do not recommend submitting stories when your cognitive functions are impaired. Friends don't let friends sub drunk. That should be fairly obvious. But perhaps less obvious is this: "I'm coming down with the flu, and have a fever, so I'd best hurry up and get all these stories submitted before I'm completely useless," is a bad idea. I say this from personal experience.
Be polite and businesslike.
As an editor, there's certain pieces of information I want in a cover letter. I want your name, the title of your story, and the word count. I also want to be informed if it is submitted as a reprint. Why? Because the first thing we do is put that information in a spreadsheet and download the story to read as soon as we have a chance. We read stories in order of receipt, so the spreadsheet acts as a queue as well. And then, we'd like to see a short paragraph about you. Three sentences, maybe. Do you want feedback? If you do, say so, keeping in mind that when you do, you're warranting that you are actually interested in receiving constructive criticism, so don't get pissed off if an editor tells you something that you don't want to hear.
Unless an editor explicitly requests it, don't tell us what your story is about. This is something you do when querying your novel, where the story runs 60k-200k words long. Short stories are a different beast altogether. (And they are shorter.) If I can't figure out what the story is about from reading the story, the story has failed.
And that's it.
Anything more is gratuitous and distracting. Perhaps even pretentious. It does nothing to help sell your story, and every additional sentence is an opportunity to shoot yourself in the foot.
Remember, when one editor turns to another and says, "Would it be wrong to reject the story and publish the cover letter?" you're doing it wrong.