Although this is a rather small ship within the fandom, and the pair have a few moments in the canon game.
In chapter three, Himiko is grieving over the deaths of her friends, Angie Yonaga and Tenko Chabashira. However, she refuses to be open with her emotions and instead bottles them all in. This is when Kokichi can tell she is lying to herself, and calls her out on it. He tells her its bad to lie about your emotions and she should be true to herself. Because of this, Himiko allows herself to cry over her deceased friends and in later chapters of the game, thanks to Kokichi and Tenko's advice, Himiko is a stronger person and is more expressive with her emotions.
In chapter five's trial, Kokichi has passed away, but before he died he left Kaito Momota notes, telling him how to act around the rest of the characters. Kaito following the notes will tell at some point that he likes Himiko, causing her to blush. However, Kaito also admits that he has ad-libbed parts of the script, and so it is unknown if this was truly part of the originial or if he just added it in. Furthermore, Kokichi is known to tell people he likes/loves them as a joke, and given he is dead at this point, it is impossible to know if he was being serious.
Both in the demo and in chapter five Himiko offers to strip Kokichi “butt naked,” the first time to make sure he wasn’t suspicious and the second time just to see him “cover his crotch”. The fact she randomly brings this up on her own suggests she may think about him often in a romantic manner. During chapter five when Kokichi took Kaito to the hangar and the remaining characters are thinking of a plan to get Kaito back, Maki Harukawa says she will kill Kokichi. Both Shuichi Saihara and Himiko tell her not to do this, Himiko going as far as saying they will be upset if they kill him. This suggests Himiko doesn’t want Kokichi to be hurt.
The ship sailed mostly as a result of a line during the fifth Class Trial when Kaito, pretending to be Kokichi in an Exisal, tells Himiko that he likes her after she tells him she doesn't even like him. Himiko easily buys this and later tells him not to bully the person he likes, referring to herself..
- Their podiums are next to each other's during the Class Trials in New Danganronpa V3: Killing Harmony.