If possible, it is generally desirable to attend Liturgy or another church service on your name day. Another tradition is for the family to include the saint's troparion and/or kontakion in the family's prayers. The icon of the saint is displayed prominently, perhaps even brought to church for all to venerate.
Often, gifts are given to the person whose name day is being celebrated, as this day is seen to be more important than a (secular) birthday.
In at least the Greek and Romanian tradition, celebrations often include an open house.