The correct choice is the first one:

> "He decides to take revenge on her."

The others sound strange and incorrect. 

I don't think there is any rule involved here except that prepositions are closely tied to verbs in phrases with specific meanings. You can't change the preposition without changing the meaning of the phrase. The other two phrases have no recognized meaning:

> "take revenge of someone" 

> "take revenge from someone"

Even the synonym "vengeance" uses the same preposition. You would also say

> "take vengeance on someone"

However, there's a phrase using "revenge" with "for" that means something different:

> "take revenge for something" or "take revenge on someone for something"