In Halacha, is it possible to claim (money) if a neighbour's noise disturbs my peace? (This would be a last resort if more friendly approaches did not resolve the problem.)
The Mishnah says:
If a resident wants to open a store in his courtyard, his neighbor can protest to prevent him from doing so and say to him: I am unable to sleep due to the sound of people entering the store and the sound of people exiting. But one may fashion utensils in his house and go out and sell them in the market, despite the fact that he is not allowed to set up a store in the courtyard, and the neighbor cannot protest against him doing so and say to him: I am unable to sleep due to the sound of the hammer you use to fashion utensils, nor can he say: I cannot sleep due to the sound of the mill that you use to grind, nor can he say: I cannot sleep due to the sound of the children. It is permitted for one to make reasonable use of his own home. [Bava Batra 2:3]
Whether you are entitled to monetary compensation would have to be decided by a bet din.
If the noise is due to normal activity, you cannot complain about it. The Chazon Ish [Choshen Mishpat 13:11] ruled, in a specific case, that the noise of a baby waking up at night is not something the neighbor who complained about it can demand be stopped.