Is there a term for "midnight" that is like "noon"

"noon" is the term for the middle of the day, round about 12.00 to 13.00, and "midnight" is from 24.00 to 1.00, at night. "midnight" is just basically a prefix added to "night", whereas "noon" is a completely different word.

Why is it "midnight", and not some term like "noon"? Is there actually a term for "midnight", that is the equivalent of "noon"?

While there is not a one-word term for midnight similar to noon, there are several poetic phrases for the middle of the night such as dead of night and hush of night. And from Longfellow's "The Goblet of Life", we have the interesting noonday night:

And also these fun phrases from "The Two Rivers":

And of course there's this from Shakespeare's Hamlet:

My Oxford American Writer's Thesaurus says the antonym for midnight is "midday" as you said.

It might interest you to know the etymology for Noon is this one:

Old English nōn [the ninth hour from sunrise, i.e., approximately 3 p.m.,] from Latin nona (hora).

Now the term indicates "twelve o'clock", but the original term used to refer to a completely different hour, so basically "early afternoon", as you can see from that etymology (that usage is classified as obsolete by the Oxford English Dictionary).

Midnight is the equivalent for noon: they both refer to a specific moment in time, and the times very close to them.