Visiting someone in hospital clearly involves seeing that person. But dancing? Does 舞 have some other, relevant meaning? Did it pertain to a traditional, sacred dance for health? Or is this ateji?
まう has a core meaning of "turn around and around". This word is actually the root of the modern verb 回【まわ】る, and in the compound 見【み】舞【ま】う and its derivational noun form 見【み】舞【ま】い, it's the older "turn around" sense that's key -- not the "dance" sense.
My copy of Shogakukan's 国語大辞典 lists this as the first definition for 見【み】舞【ま】う (emphasis mine):
Note those two verbs on the end of the definition -- they both include the idea of "going around [to see someone / something]". That's where the まう verb comes in.
In addition, みまう can be spelled 見【み】廻【ま】う, using the 廻 kanji that can also be used to spell 廻【まわ】る.