according to some “scholars” (that is how they are called, I know nothing about them or their scholarship) the custom of gift giving on the 25th of december is done as a zecher for gifts given on the december 25th in question, namely the gift of gold (chanuka gelt?), frankincense and myrrh to a certain baby by three men who may or may not have been wise. over the years this “zecher” has been adapted by different cultures so that the original zecher has been completely corrupted (unless there is an opinion out there that can explain the connection between the latest iphone, tickle me elmo or video game) to the point where there is little religious significance to the present day gift giving.
Whether or not one should engage in the practice of gift giving on chanuka, is something I will leave to each individual to ask of their own Rav.