I think the answers here have covered a lot of the territory.
However, consider "(old) chestnut", which can have a negative connotation, for example in describing a well-worn joke or story, or a neutral or positive connotation, for example in describing a familiar aphorism or saying.
Oh no – Grandpa saw a pack of my business cards, and added me to his "hilarious" e-mail list. His messages are full of internet meme chestnuts from 2007.
In this sentence, the negative connotations are clear. The words "internet meme" provide clarity of context (if that is indeed the context you want), but may not be necessary, depending perhaps on your audience or further context provided elsewhere in your text.
Here are some links that you can follow to confirm the definition and suitability (note the definition that Google provides atop the search results):
- Wiktionary (old chestnut)
1. (idiomatic) A well-worn story.
- Google (old chestnut)
1. a joke or story that has become tedious because of its age and constant repetition.
- Dictionary.com (chestnut)
6. an old or stale joke, anecdote, etc.
- Thesaurus.com (joke; provides chestnut)
Here's the Wiktionary etymology information, which, along with those links, suggest that "chestnut" is very close to exactly what you're looking for:
Originally as chestnut, with "old" for emphasis. Popularized US 1880s, particularly Northeast and Midwest, with various theories propounded.
A commonly cited theory, viewed by the Oxford English Dictionary as "plausible" and cited by Brewer’s, is that it was coined by Boston comedic William Warren Jr., quoting from 1816 English melodrama The Broken Sword by William Dimond. One of the characters in the play is a boor, and when once recounting a tale mentions a cork tree, which is corrected by the character Pablo as "A chestnut. I have heard you tell the tale these 27 times." This line was then apparently quoted at a dinner party by Warren in response to a boor there, and proved popular. Note that William Warren Sr. had previously played Pablo on stage, but died in 1832, so the phrase was presumably popularized by the son, William Warren Jr.
Indeed, you've asked for the English equivalent of "баян". According to its Wiktionary entry, "баян" – literally, "accordion" – gets its figurative meaning of "old joke" from an oft-repeated and well-worn joke about an accordion (or rather, two accordions). With this ideational correspondence in their origins, "chestnut" even seems to have a similar flavor in English to "баян" in Russian, even if not the same currency on the internet.
- Wiktionary (баян)
The “old joke” sense originated from a joke (Internet meme) at http://www.anekdot.ru — "Хоронили тёщу — порвали два баяна" (when we buried my mother-in-law, we broke two accordions).
(I would even go so far as to suggest that someone should edit the Wiktionary entry for "баян" to include "chestnut", but only as I gather from my research as set out above and from reading the comments on this page. To be clear, I don't speak any Russian and I have no idea what human burial has to do with accordions, or breaking them. Edit: I developed a hunch that accordion wreckage was the measure of a really good Russian party. That is confirmed here.)