Am I allowed to give a gift to my wife while she’s an aval ?
After the shloishem?
Yes, after shloshim. The gift should be something not so significant and therefore does not cause excessive happiness.
The Mishnah Berurah (ibid. 21) quotes this ruling, though he brings it from Magen Avraham (who explains differently), which makes his opinion a little unclear (seeDivrei 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.
Therefore, under circumstances where the gift is “required” as part of a relationship, for instance from husband to wife, there is room for leniency (see a similar ruling in Nitei Gavriel, Aveilus Vol. 2, Chap. 14, no. 13).