"Kid" is not capitalized. "The kid was silenced by his mom" is correct. "The kid was silenced for making noise" is also correct but might sound better if you say "The kid was silenced for making too much noise"

Thank you so much, Dave

Just a note that "silenced" sounds a little harsh to me. Maybe that's your intention. You can also silence a phone or silence a political opponent, for example. But maybe "quieted", or some other phrase could sound better in this case.