We (Greek tradition) celebrate them like birthdays - cake and presents for the kids, and greetings from family and friends.
We go to Church if possible - generally, Greek priests don't work a day job, so they celebrate Liturgy during the week if asked, especially for a common nameday or important Saint's feast day. Kyriaki is a common Greek name, so we ALWAYS have a liturgy for her day. And my priest makes the same corny joke about seeing so many Kyriakis (Sundays) on a Monday or whatever day of the week it is. Next year it will fall on a Sunday, so we'll get some joke about it being doubly our nameday or something!
Everyone who can receives Communion on their nameday - this is more of a big deal for those who receive infrequently, but it's still something we try really hard to do even if it means missing the morning at work or study.
Most of the customs revolve around giving to others - once we're no longer children, it's more about us giving to others rather than being given gifts. Traditionally the custom is to give to the poor, which often works out to hosting a feast or at least bringing food to Church for everyone to share. Another common tradition is to bring the bread and wine for an artoklasia, which I forget the russian name for. Litya? The five loaves of bread blessed by the priest, generally at Vespers. I am determined to make the loaves one year, but I've always been too sick to do so! One day.
A non-Church related custom is that if it's your nameday and you're drinking with friends, you pay, since that's giving to the poor...
My thing is flowers - I don't have the energy to bake, or a lot of money, but I always bring flowers for St Kyriaki and the Church. I don't know if she likes gerbera daisies but I do, so that's what I bring! This year I went to a flower farm about 15 minutes from where I live and asked them to make me up some arrangements, so I have a big one for Church and a smaller one for our living room icon corner table (yes, my protestant family has an icon corner, and my Mum's actually quite proud of it!) They do beautiful arrangements, and I did three kinds of flowers this year and they added greenery. Gerberas (joy and purity), lilies (purity and spiritual maturity) and purple asters (pure love and patient endurance), which seemed perfect for a virgin great martyr. Here's pictures, because I can't resist - today's my nameday (7th of July):
Church flowers:
Home flowers: