May one give a gift in the year of aveilus and write that the gift is from the family, do you have to specify that it’s not from the aveil?
The answer to this question depends on who the principle “giver” of the gift is. If the principle giver is the aveil, and the word “family” is just a veil for the aveil, then using the word “family” would not be sufficient to permit the gift, and the gift would have to be explicitly given by a different member(s) of the family. However, if the gift is not primarily from the aveil, but rather from all members of the family (or primarily from another member of the family), then it would be permitted. The fact that one member of the family is an aveil does not prohibit the “family” from giving gifts.
Sources: The prohibition of giving gifts in aveilus is on account of the issue of she’elas shalom, or, alternatively, on account of the joy it generates (she’elas shalom, according to several authorities, is not prohibited for the entire year, whereas many prohibit giving gifts during the entire year, suggesting a different rationale; see Divrei Sofrim, Aveilus, siman 385, Birur Halachah, s.v. ve’assur lishloach). According to both possibilities, the family is not prohibited from giving gifts. However, if the main “giver” is clearly the aveil, using the family as a guise would not solve the issues.