In Chapter 5 of Danganronpa: Trigger Happy Havoc, after Makoto was saved by Alter Ego before getting executed by Monokuma, he was thrown to a dumpster that was a room in where Monokuma stored trash. Kyoko then arrives to save Makoto by bringing him some food, falling between the trash. After getting up of there, Naegi points out that Kyoko had "garbage" on her head, which was a half-empty cup noodle. Kirgiri then throws it away and continues with her rescuing mission. The scene resulted in Kyoko having a few canon sprites with cup noodle on her head.
Kyoko seems to hate cup noodle with a "buring passion" because of a certain event that happened in the School Mode during the Dining Hall Trip. In the following event, Naegi is hanging out with Kirigiri to spend time with her. If the player chooses to ask her if she liked cup ramen/noodle, she will be a bit shocked about the question, then she will start feeling "ill" because of Makoto's comment, and so he'll notice that things didn't go well.
In the ninth episode of Future Arc, an empty ramen cup is seen among the things that Kyoko had eaten with Makoto, Aoi Asahina, and Ryota Mitarai. The incident from Chapter 5 reappears in the tenth episode of the same arc in Makoto's flashback of his moments with Kyoko.
Like other cargo ships from the franchise, Kyoko x Cup Noodle/Ramen received popularity in the moment she had a cup of noodles on her head during her rescue mission, which even gave her a couple of canon sprites wearing it. As a result of said moment and despite Kirigiri's supposed hate of cup noodles stated in School Mode, she's seen in canon artworks with a cup of noodles/ramen on her head, with this happening unknowingly or her just not minding it. This inspired fans to also draw pictures portraying her eating ramen/noodles or with her with a cup ramen/noodles on her head like in this image, that shows her eating the product while in this other image Kirigiri is seen with a cup of ramen/noodles on her head, not minding it, or in this other one where is seen Kirigiri inside a cup of noodles/ramen. A fan also created a video about Danganronpa's characters like Makoto, Kyoko, or Junko Enoshima consuming the cup noodles.
With the popularity that the cargoship gained, there are different products like Kirigiri's model with the product on her head that was made by a fan, that is still in store. Thanks to this, there are other people who made similar models involving her and the product, like this one, and this other one. A fan also created a chibi Kirigiri inside a cup of noodles/ramen, while other fan created a keychain and finally, these fans cosplaying Kirigiri with, again, the product on her head.
Due to the amount of both canon and fanon content of Kyoko wearing cup noodle on her head, Kyoko and cup noodles went on to become one of the biggest memes of Danganronpa franchise. Despite the ship's popularity, it has its respective popular ships that overshadow the ship itself, like Naegiri and Celesgiri; just like in any other cargoship's case, such as Aoi and doughnuts (overshadowed by Sakuraoi) or Mondo with butter (overshadowed by him with Kiyotaka and with Chihiro).