Precociousness is the noun: the quality of being ahead of your age. You use it to name the thing itself rather than describe a person — 'his precociousness surprised his teachers', 'there was a precociousness about the way she argued'. It covers both sides of the adjective. It can mean early ability, which is a compliment, and it can mean an early-adult manner, which can be a gentle complaint. Which one you mean usually comes from the sentence around it. Worth knowing: there is a second noun from the same adjective, 'precocity', and that is the more common one in careful and literary writing — 'musical precocity', 'the precocity of the child'. 'Precociousness' is perfectly correct and slightly plainer and more everyday-sounding; some writers feel it is the clumsier of the two because it is longer. Both are fine, and English does this often, keeping a Latin-shaped noun and an English -ness noun side by side. Neither is common in speech; a parent talking about their own child would just say 'she's very advanced for her age'.