I know how to wish a happy birthday in Latin: Bonum diem natalem! (There are other options as well.) It just occurred to me that I do not recall coming across any ancient birthday congratulations. Do we know how the Romans wished a happy birthday? I am looking for attested expressions from classical literature.
optime natalis! / best of birthdays!
from Ovid, Tristia, 5.5, line 13?
vivat … consumatque annos … suos / long life to her … and may she pass to the end of her years
also from Ovid, Tristia, 5.5, line 23.
Other possibilities from genethliacon/birthday poems include:
transeat hic sine nube dies / may this day pass without a cloud
Propertius, Elegies, 3.10, line 5 (I have actually used this line in a birthday greeting to a Classicist)
at tu, Natalis, multos celebrande per annos, candidior semper candidiorque veni / But you, Birthday, come to your honours for many a year, come ever brighter and brighter still
Tibullus, Elegies, 1.7, lines 63-64
dicamus bona verba: venit Natalis ad aras / let us speak only good words: a Birthday comes to the altar
Also Tibullus, Elegies, 2.2, line 1
My guess would be, there was no special or particular phrase, or if there was it didn't survive.
The closest I found to a direct birthday greeting was Pliny Ep. 6.30, a birthday letter to his friend Fabatus. But he never directly wishes him a happy birthday.
Martial 8.64 (which doesn't seem to be available in Perseus?) talks about the custom of gift-giving, and Ovid's Tristia 5.5 and 3.13 mention the religious rituals, but none mention any special or particular greeting.
Plautus' Captīvī I.2 contains this exchange:
Erg.: Quia mi est natalis dies / propterea te vocari ad te ad cenam volo.
Heg.: Facete dictum.
Erg.: Because it's my birthday, so I would like for you to be invited to dinner with me.
Heg.: Politely said.
But this seems to be a refusal of his invitation, nothing more.
Here's the only classical-age birthday wish I've been able to find on PHI:
C. Plinius Traijānō imperātōrī
Optō, domine, et hunc nātālem et plūrimōs aliōs quam fēlīcissimōs agās... ('I wish, Sir, that this birthday, as well as many others to come, bring you much happiness...' Plin.Sec. Ep. 10.88.1)
I suspect there exist more in the wider corpus of pre-carolingian Latin, but I haven't looked for them. This formula - fēlīcem nātālem agās - looks to me to be a safe and idiomatic bet, but if you want some variety, there's also the word nātālicius (diēs), as well as grātulārī, which can be used as follows:
grātulārī: alicuj dē victōriā/in hōc/prō ingeniō tālī/victōriae tuae/cum aliquid factum est/quod aliquid factum est/aliquid factum esse/eam rem
Studies of the Vindolanda Tablets (No# 291) indicate a linkage between "dies natalis" and "sollemnis" (solemn, ceremonial; traditional, customary). A letter to Sulpicia Lepidina, from Claudia Severa, wife of Aelius Brocchus, inviting Lepidina to visit for Severa's birthday.
"Claudia Severa Lepidinae suae salutem, III Idus Septembres soror ad diem sollemnem natalem meum rogo, libenter facias ut venias ad nos iucundiorem mihi."
"Claudia Severa to her Lepidina, greetings. On 11th September, sister, for the day of the celebration of my birthday. I give you a warm invitation to make sure that you come to us."
Supported by refs: Fronto ad.Ant.imp 1.2: "...te mihi ab dis die tibi sollemnissimo natali meo precatum."
"...you, having prayed to the gods, for me, on this my birthday, on this most traditional day, for you.";
Horace Od. 4.11:
"iure sollemnis mihi sanctiorque,
paene natali proprio,"
"Deservedly almost a (day) of reverence for me with my own birthday."
You would have to start with this woeful line of poetry by Cicero:
O fortunatam natam me consule Romam.
with this translation by JPOMorford:
O happy State for your natal date and my Consulate.
Mary Beard quoted this line in an acute review of Cicero in the London Review of Books
one of the worst pieces of Latin doggerel to have made it through the Dark Ages (‘O fortunatam natam me consule Romam’ – a jingle with something of the ring of ‘Rome was born a lucky city, when I as Consul wrote this ditty’).
But since Consulship doesn't come into it, and it's te not me, here's: "O Happy world that you were born."
O fortunatum orbem nato te. ../ nata te for a girl
To be more authentic, the celebration of a birthday would involve natalitia (starts with 3 long syllables) birthday sacrifices: Do, or Damus "I give /we give," or the subjunctive would say that we take part in this ceelbration
A modern Latin source which may be of interest has three versions of Birthday Salutations.
Winnie Ille Pu - A.A.Milnei(in Lat conv. Alexandro Lenard, ?1960)
BVBO (Owl or Wol)
Ita Bubo scripsit . . . et ecce inscriptio: 'FLICM FELCM NTAALM TATALM NATATALM.'
' Utinam dies iste saepe et feliciter revertatur,' dixit Porcellus, appropinquans.
Scriptum legimus: felicem natalem amanter ominatur Pu.