Cutting Nails for a Child During the Thirty Days of Mourning

Is it permissible for a child to cut nails during the thirty days of mourning for his father?

It is permitted. See the source.

In the Gemara Moed Katan, page 14: "Shmuel said: A child born during the festival may be groomed during the festival, as there is no greater imprisonment than this. During the festival - yes, initially - no. Rabbi Pinchas objected: All those who are permitted to groom during the festival are permitted to groom during the days of mourning. Those who are forbidden to groom during the festival are forbidden to groom during the days of mourning. And if you say there is a dispute regarding the child, it implies that mourning applies to the child. But it was taught: they tear the garment for a child due to emotional distress! - Rav Ashi said: Does it state that it is forbidden? Perhaps some are forbidden, and some are permitted. Ameimar, or if you prefer, Rav Shisha, son of Rav Idi, taught thus: Shmuel said: A child may be groomed during the festival, whether born during the festival or initially. Rav Pinchas said: We also learned: All those who are permitted to groom during the festival are permitted to groom during the days of mourning. Those who are forbidden to groom during the festival are forbidden to groom during the days of mourning. If you say a child is forbidden, it implies mourning applies to the child. But it was taught: they tear the garment for a child due to emotional distress. - Rav Ashi said: Does it state that it is forbidden? Perhaps some are forbidden, and some are permitted." It is explained that there is no prohibition to groom the child in mourning, and they tear the garment only due to emotional distress. This is also ruled by the Beit Yosef in Siman 389 in the name of "Halachot Smachot," and similarly in the Mordechai in "Hilchot Avel" at the end of Siman 495. And so it applies to the prohibition of cutting nails. The rabbis are divided on whether there is education in mourning for a child. According to the opinion of Ri"tz Giat and T"Z and B"H, there is a law of education in mourning for a child. They hold that the law explained in the Gemara, that they tear due to emotional distress, applies to a child who has not reached the age of education, but for a child who has reached the age of education, they tear by law. However, in Sh"K Siman 384, S"K 5, it is written that there is no law of mourning for a child, and so it is written in "Beit Hillel" in Siman 389. Practically, the main law is followed, that there is no education for a child in mourning.