"Cyclops Pony" is the name given to a bluish pony within the background of the My Little Pony episode "The Best Night Ever" where due to an unfinished animation error had shown to the audience a quick flash of the pony showing only one eye within the middle of two closed eye lids.
The pony was then used for various amounts of fan-fiction artworks and video and is similar to other ponies such as 'Derpy Hooves' who was named Derpy after an animation error with her eyes making a Derp face.

Episode Scene Summary (MLP Wikipedia)

"Once Applejack, Fluttershy, and Rainbow Dash meet up with the rest of the ponies, Twilight casts her latest spell, transforming an apple Spike places on the ground into a carriage to take them to the Gala. She proceeds to turn a group of four mice that Fluttershy had brought with her at Twilight's request into horses to pull the carriage. The spells are a near perfect success, except the horses' faces still had mouse whiskers. Thanks to Opalescence, however, the "horses" are scared away. With some eyelash batting, Rarity convinces two of her stallion neighbors, Caramel and Lucky, to pull the carriage instead." (The last bit is when we see the Cyclops Pony)

(shown at 1:35 of the video)

Fan-Made Images and Videos [researching]

The spread of the cyclops pony was critical within communitys such as a FaceBook page and several videos showing the character being reacted by others within images and videos