Are there different ways to say good morning in Latin?
Would bene mane be okay?
I myself have wondered about this on numerous occasions, and here's what I've learned:
There were no separate greetings for different times of day in ancient Rome, unlike in modern European languages. The universal greeting salvē "be well!", or the more bombastic havĕ (from Punic ḥave "live!" - don't get fooled by the common spelling avē), or the more informal salvus sīs were used universally. I might speculate that the reason for this is that modern greetings reflect a business-oriented world and its belief in the role of chance in the outcome of a successful business-day, while the only chance thing the Romans worried about was their health - for the rest they had their reciprocal relations with the gods (fidēs X religiō).
Less speculatively, the Roman day always began at sunrise and ended at sundown, because every daylight hour had to be maximised in a lightbulbless world (recall even the modern daylight saving time) - and therefore subject to seasonal variation depending on the location, suggesting a smaller fixity of time in general. Somehow connected with this is the fact that there's no noun corresponding to our "morning" in Latin: māne is more of an adverb "early, in the morning, to-morrow", although it can be used as a noun in the three cases where it resembles one (nom/acc/abl). The cumbersome tempus mātūtīnum "the morning-hours" is the closest thing that exists otherwise, eventually shortened to mătŭtīnum > *mattīnum (Italian mattino), and there's also a decent selection of terms for the more narrow meaning "dawn".
In Medieval & New Latin one can find expressions like:
I'm confident these would have seemed like awkard calques from Greek to the ancient Romans, fit perhaps for use in a letter; in the case of the last selection, a letter to the Emperor would be fitting.
Bene māne means "nice and early in the morning", "at the crack of dawn", perhaps even before it. Cute factoid: māne itself comes from a PIE root meaning "good, timely", also reflected in Mānēs "the good ancestral spirits".
Bonum māne, a phrase with a pedigree of some centuries that you can find as a suggestion on plenty of generic translation webistes, would probably be as understandable as "the good tomorrow"; albeit clārum māne fenestrās intrat is actually used by Persius, figuratively I think, as "the bright tomorrow/early hour enters the window".
In fact, it just struck me that even the Medieval examples I offered use the word diēs, and then yet another explanation dawned on me. It's does find some (poetic) use in the meaning "daylight", and was originally used to mean "day sky", and its personification was none other than Jūpiter < Djou(s) Patēr, "Sky Father", Ζεύς, which was transparent to the speakers as evidenced by it reshaping as Diēs Pater. This brings to mind expressions like Juppiter tibi īrātus sit! "May the Sky Father be angry with you!" (a curse), malus Juppiter "unkind Sky Father" (of northern weather), and conversely all those epithets like Optumus Maxumus: in fact you can see on Logeion (on the right) that bonus is the second most common collocation with this name. This suggests that an expression like bonus diēs was likely to be interpreted as a reference to the Sky Father's kindness/favour, perhaps poetically or as a minced oath (in an exclamation), and hence was unlikely to be used like we would be inclined to use it.
I too find myself reaching for these time of day-specific phrases, and these are my attempts at an explanation of why they don't exist - or as far one can surmise, didn't in Classical times. I imagine they must have appeared by the 5th century at the latest.
Reading Plautus is a great way to learn colloquial Latin. All my examples below are from plays such as Miles Gloriosus. The Romans did not say "good morning".
Heus! is what you say when you call out to somebody Hello!
Quid agis? is a normal casual greeting, "How are you doing?"
Quid fit? is another casual greeting, "How goes it?"
Salve is the standard courteous greeting, "Be well."
Salva sis or Et tu, salve is the standard courteous response, "Health to you as well."