Can a father give his children (older) a small birthday gift even though they are avelim toch yud beis chodesh, where normally they should not be given gifts? Perhaps a father is not considered she’elas sholom?
After the thirty days (sloshim) it is permitted to buy a modest birthday present.
Sources: The regular sustenance of children is not, of course, considered a “gift” and included in she’elas shalom (Mahril, no. 31, which is the source for the prohibition of giving gifts to an aveil (Rema, Yoreh De’ah 385:3), writes that the issues is she’elas shalom), but it is a little hard to extend this to special gifts that a father gives his children.
However, there is room for leniency based on the Taz (Orach Chaim, 696:3) who writes that because the custom today is not to be particular about she’elas shalom after shloshim yom, it would also be permitted to give gifts. [The ruling is not entirely clear, and some have asked (see Eliyah Rabbah) that the Rema never states that today’s custom is to be lenient concerning she’elas shalom, but only that the custom in some places is to be lenient.]
Mishnah Berurah (ibid. 21) quotes this ruling, though he brings it from Magen Avraham (who explains differently), which makes his opinion a little unclear (see Divrei Sofrim, aveilus, Eimek Davar 47); he adds, however, that the gift should not be something that causes simchah. The same ruling is given by Derech Chaim.
Based on the principle that the halachah follows the lenient opinion in aveilus (see Shiurei Berachah, 374, who writes that he has a mesores from all morei horaa that this applies even to disputes between poskim), and in view of the need involved, it would be permitted to mark birthdays with a modest present — not perhaps as grand as those of other years, which would be a “gift of simchah,” but something, nonetheless.