Although we know Romans celebrated birthdays, we don't know if they wished one another the exact phrase "Happy Birthday!" But that doesn't mean we can't use the Latin language to wish someone a happy birthday. The following seems to be the best way to express "happy birthday" in Latin.
Felix sit natalis dies!
Using the accusative case, specifically the accusative of exclamation, felix sit natalis dies is one way to say "happy birthday." Similarly, you could also say felicem diem natalem.
Habeas felicitatem in die natus es!
Habeas felicitatem in die natus es is another possibility. The phrase roughly translates to "on happiness to love you."
A third way to wish happy birthday is Natalis laetus mihi! if you want to say "happy birthday to me." Or, Natalis laetus tibi! if you want to say "happy birthday to you."
Celebrating in Ancient Rome
The ancient Romans observed different types of birthday celebrations or dies natales in Latin. Privately, Roman men and women marked their own birthdays and the births of family members and friends with gift-giving and banquets. Fathers gave presents to their children, brothers gave presents to sisters, and slaves gave presents to their master's children.
One custom was to celebrate not on the specific date an individual was born but rather on the first of the month (calends) in which the individual was born, or the first of the next month.
Gifts given on birthdays include jewelry; the poet Juvenal mentions parasols and amber as gifts, and Martial suggests togas and military clothing would be appropriate. Birthday feasts might have entertainment furnished by dancers and singers. Wine, flowers, incense, and cakes were part of such celebrations.
The most important feature of Roman personal birthday celebrations was a sacrifice to the genius of the housefather and the juno of the housemother. The genius and juno were clan symbols, representing a person's patron saint or guardian angel, who guided the individual throughout life. Genii was a sort of middle power or intermediary between men and gods, and it was important that votive offerings be given to the genius each year in hopes that the protection would continue.
People also held similar celebrations for the birthdays of close friends and patrons. There is a wide variety of elegies, poems, and inscriptions commemorating such events. For example, in 238 CE, the grammarian Censorinus wrote "De Die Natali" as a birthday gift for his patron, Quintus Caerellius. In it he stated,
"But while other men honor only their own birthdays, yet I am bound every year by a double duty as regards this religious observance; for since it is from you and your friendship that I receive esteem, position, honor, and assistance, and in fact all the rewards of life, I consider it a sin if I celebrate your day, which brought you forth into this world for me, any less carefully than my own. For my own birthday gave me life, but yours has brought me the enjoyment and the rewards of life."
Emperors, Cults, Temples, and Cities
The word natali also refers to anniversary celebrations of the founding of temples, cities, and cults. Beginning with the Principate, Romans also celebrated the birthdays of past and present emperors, and members of the imperial family, as well as their ascension days, marked as natales imperii.
People would also combine celebrations: a banquet could mark the dedication of an association's banqueting hall, commemorating an important occasion in the life of the association. The Corpus Inscriptionum Latinarum includes an inscription from a woman who donated 200 sesterces so that a local association would hold a banquet on her son's birthday.
Argetsinger, Kathryn. "Birthday Rituals: Friends and Patrons in Roman Poetry and Cult." Classical Antiquity 11.2 (1992): 175-93. Print.
Ascough, Richard S. "Forms of Commensality in Greco-Roman Associations." The Classical World 102.1 (2008): 33-45. Print.
Bowerman, Helen C. "The Birthday as a Commonplace of Roman Elegy." The Classical Journal 12.5 (1917): 310-18. Print.
Lucas, Hans. "Martial's Kalendae Nataliciae." The Classical Quarterly 32.1 (1938): 5-6. Print.