Example Memes: (Searching these will render serious results)
Pedobear
RaptorJesus
Happy Negro
Anonymous
Of course BYOND has a few "mini-memes"...But I'm not going to say them for fear of persecution...but I'll post the ones that I agree with..So feel free to take a stab in the dark at what mascots/memorable phrases you think BYOND has to offer.
Current BYOND Memes: *Don't Forget The Replies!*

The Bunchie While this Phallus shaped creature didn't originate on BYOND, it's definitely made quite an impact on the BYOND community, both having its own fan club and having appeared well over 100 times it's truly a BYOND meme.
"well, i do plenty fine with women. Plus i get very high grades in school. So i dont know why i care what you think."
And I bet you thought you had heard the end of this phrase...It's been repeated over 100 times thanks to Propaganda.

The Evil RSC Extractor Known to eat games...especially those of naruto origin.

Nadrew's Required Organs So thats where they all went...

The Ever-Repetitive Pbag It will cause you to instantly achieve rip-status.

The Blue Guy BYOND's official mascot...We guess...While it is unsure if he has a name he was thought up on a trip to India...probably while smoking some serious Indian herb.

The Narutard
Established in 1999, perfected in 2007.
No one on 4chan devised the term, and your description isn't very good as to what they actually are.