Feelings/emotions are expressed here with the Dativ (mir ist), not with Nominativ (ich bin).
You could argue, that this is done for differentiation purposes, because (as Matthias mentioned) "Mir ist übel" and "Ich bin übel" mean different things. But that does not explain other uses like:
Mir ist, als hätte es geklopft. (I think someone knocked at the door.)
You are expressing a feeling, you are uncertain if really someone knocked. An "Ich bin" construction does not exist here.
For further readings (and if you want to lobotomize yourself) I recommend "Die Kodierung von Emotionen in Texten".
One possible explanation how this evolved is the shortening of sentences. Let us take as example:
Es ist mir kalt. (It's cold here for me.)
Now we have a typical sentence with a dative object. It could be that the object was transferred to the subject to shorten the sentence, because people are lazy (sprechfaul). Then we get:
Mir ist kalt. (I am cold.)
But this is guesswork. I have no reference to back this up.