Just as the Sages forbade engage in commerce on Shabbat, they similarly enacted a provision forbidding giving gifts, which resembles transactions. Even though no money is paid when a gift is given, it is nevertheless forbidden on Shabbat because it resembles a commercial transaction.
This is an issue that is quite common, such as in the case of a husband who wishes to give his wife a gift for her birthday or Mother’s Day at the Shabbat table, or a Rabbi who wants to present a Bar Mitzvah boy with a gift from the congregation on Shabbat in the synagogue. To circumvent this prohibition, the one giving the gift should give it to somebody else before Shabbat, who has in mind to receive it on the recipient’s behalf. For example, the husband can give the gift to his grown son before Shabbat and ask him to receive it on his mother’s behalf. The Halachic principle of "Zachin Le’adam She’lo Be’fanav" establishes that one can perform a beneficial legal act on behalf of another person even without that person’s knowledge, and thus a person can receive a gift on behalf of another person, and it then becomes that other person’s property already at that point. And since the gift belongs to the wife already before Shabbat, the husband may give her the gift on Shabbat, as he is merely giving her something which is already hers. Similarly, the Rabbi should hand the Bar Mitzvah gift to somebody before Shabbat, instructing him to have in mind to take it on behalf of the Bar Mitzvah boy. If the Rabbi did not have somebody else acquire the gift on the boy’s behalf, then he should, if possible, instruct the boy to have in mind not to legally acquire the gift until after Shabbat.
However, Hacham Ovadia Yosef writes (in the newly-published edition of Hazon Ovadia, vol. 6, p. 14; listen to audio recording for precise citation) that in the case of a Mitzvah there is more room to be lenient in this regard. And therefore, if a Rabbi wishes to give a Torah book to a Bar Mitzvah boy, it would be permissible to simply give the gift on Shabbat. Although it is preferable to use the solutions described above, if this is not possible the boy may be given the gift on Shabbat.
Summary: One may not give a gift on Shabbat, unless before Shabbat he gives it to a third party and has him have in mind to make the acquisition on the recipient’s behalf, or if he tells the recipient to have in mind not to take legal possession until after Shabbat. In the case of a Mitzvah, such as if a Rabbi gives a Torah book to a Bar Mitzvah boy, the gift may be given, though it is preferable to utilize one of the methods described above.
