What should a cover letter for a short story contain? Do I need to tell what is it about, try to "sell" it? Should I write about myself?
Unfortunately, the magazine's website doesn't have anything useful to say. So one would imagine that standard cover letter advice would apply here. (I got some of this information from the links at the end of this answer, but have condensed, and reworded significantly.)
- Your name, contact info, and a word count
- Follow their cover letter guidelines (where they have them - in this case there are none)
- The letter should be to a specific person, when you can do that (for The New Yorker, just "Fiction Editor, The New Yorker")
- Mention the magazine, but briefly. (So that it's clear your note isn't a form letter)
- Is this a simultaneous submission? Some magazines want to know that. Some don't allow it at all. (Check their guidelines.)
- Keep it short. Magazines get a lot of submissions, and a long cover letter is unlikely to be read.
Since the New Yorker doesn't really say what they need, I'd either contact them and ask for clarification or find out what other writers have done. Do they even want a cover letter? Maybe this info should just be in the email? Or maybe they just want an email address and nothing else? Ask, or put together a very brief letter. (A short cover letter is unlikely to be a problem.)
I use the Purdue Online Writing Lab for almost every writing question I come across. You can check out what they have to say about cover letters here: https://owl.english.purdue.edu/owl/resource/723/03/
They have this to say about the argument portion of your letter:
"Your argument is an important part of your cover letter, because it allows you to persuade your reader why you are a good fit for the company and the job. Carefully choose what to include in your argument. You want your argument to be as powerful as possible, but it shouldn't cloud your main points by including excessive or irrelevant details about your past. In addition, use your resume (and refer to it) as the source of "data" you will use and expand on in your cover letter."
Since you're supposed to be persuading the recipient why you're a good fit for whatever position you're applying for, I suggest doing a little bit of both of your suggestions. Give some good teasers from your story that will make your characters and story lines sound desirable.
But most importantly, you definitely need to write about yourself. You're selling yourself with this letter, not just a short story.