Actually every day is a special day for prayer, but I am not aware of any special-ness about one's Hebrew birthday for praying or of any special prayers to say on that day. A lot of people I know sing "Happy Birthday to You!..." and I think it makes the person happy if the person doesn't object from embarrassment or some personal reason.
The Jerusalem Talmud indicates that it is a “lucky day” for a person. When the Amalekites attacked the Jewish people after the Exodus, they chose soldiers whose birthday was on the day of the battle. They perceived that a person's birthday is a lucky day for him, and therefore he will be successful in battle.
The Ben Ish Chai (Rabbi Yosef Chaim of Baghdad) writes that some people celebrate their birthday because the day is a good sign for that person. He personally celebrated birthdays in his home. Rabbi Yisrael Lifshitz (author of Tiferet Yisrael, commentary on the Mishna) instructed his children that when one of them has a birthday the others should visit and bless him. Similarly, distinguished members of Jerusalem's Jewish community used to visit Rabbi Shmuel Salant on his birthday and offer him their blessings.
Others emphasize the more serious side of birthdays. Rabbi Avraham Binyamin Sofer (author of Ktav Sofer responsa) used to sequester himself on his birthday and 'soul-search.' The day a person is born he receives the most precious gift of all - Life! Therefore, it is a day for introspection, a day for asking, "Am I using this gift to its utmost potential?"
