What is the possessive noun in the sentence the telephone of the children?

The possessive form for "the telephone of the children" is the children's telephone.

Note: The term "the telephone of the children" or "the children's telephone" is NOT a sentence, it's a noun phrase, a group of Words (without a verb) based on a noun (telephone) that functions as a unit in a sentence.

Example sentences:

The children's telephone needs a new battery. (subject of the sentence)

I bought a new battery for the children's telephone. (object of the preposition 'for')