The concept of a name day goes back to the Middle Ages. It started with the Orthodox and Catholic churches as a way to combat, what they felt, was a pagan celebration. Birthdays were considered to be festivals for the rich, for kings, and for flase gods. (see Natalis Invicti)
There were other theological issues as well The Lore of Birthdays is often sited for this:
The custom of birthday celebration started by the Romans religions and was developed by the Greeks. They believed that everyone had a protective spirit / daemon / angel who attended his or her birth and watched over him in life. This spirit is assigned to the birthday boy or girl by determining the God or Goddess born on the day the celebrant is born. Through birthday celebrations, these spirits will visit the venue of celebration.
In response, the Church began focusing on the correlation between a person's birth and the calendar of saints. The celebration took on religious importance, as it tied the individual with a given saint or martyr. (It's important to note that because different calendars exist in different regions and faith traditions, a name day is not always the same for two people.)
The idea of name days has never really taken hold in the US the way it has in Latin American and Eastern countries. In some places, the celebration of the name is just as important as the person's birthday.
As a religious, I'm constantly aware of the liturgical calendar and the various feasts and memorials throughout the year. Today (June 15th) is the memorial of St. Vitus, this is considered my name day. (In Serbia, Vidovdan is celebrated on the 28th.)
Personally, there's not a whole lot involved in celebrating my name day. There's a special intention for me during our communal prayers, and everyone tells me "Happy Name Day," but that's about it.
But still, there is something important that comes out of this day. I am reminded that I belong to a different kind of life. As a religious, the name day ties me to my Church and my faith, giving me opportunity to reflect on the journey I've made in my life. It reminds me that I'm part of something unique.
My name day is a celebration of being a Catholic, a Christian, and a Capuchin. There's no cake, no presents, not even a card. But it is a day that makes me happy to be alive.