The entry for Anna in Wiktionary certainly states that it derives from the Hebrew Hannah. And this is how Augustine uses it in The City of God against the Pagans, in book 17, when referring to Hannah, the mother of Samuel (mater quoque ipsa Samuelis Anna ...)
However, there is the possibility that Anna is in fact a Latin name, based on the Roman goddess Anna Perenna. Her festival took place on the Ides of March (March 15th), the beginning of the new year (Ovid, Fasti, 3.146). Macrobius notes that:
... publice et privatim ad Annam Perennam sacrificatum itur, ut annare perennareque commode liceat
... public and private sacrifice is offered to Anna Perenna, so that
we might prosperously pass the year [annare] and for many years to come
I think that this clearly links the name Anna with annus. She was the goddess of the new year, if you will.
The cult of Anna Perenna is also mentioned by Martial (Epigrams, 4.64.17).
But it is Ovid who gives us the fullest picture of her and the ribald festivities that took place in her honour in book 3 of the Fasti. Revellers, sitting in the sun, would toast Anna Perenna:
sole tamen vinoque calent annosque precantur,
quot sumant cyathos, ad
But they grow warm with sun and wine, and they pray for as many years
as they take cups, and they count the cups they drink.
Ovid, Fasti, 3.531-2
Further, among other stories surrounding her, he explicitly links Anna Perenna with Dido's sister Anna (see: Fasti, 3.543-654). After Dido's death, she flees into exile, spending some time in Aeneas' Lavinium but, due to Lavinia's jealousy, she has to leave. Eventually, finding no refuge, she was turned into a river nymph by Numicius, whereupon she was called Anna Perenna. This is all highly anecdotal, of course, but it does at least establish that the name Anna was known to the Romans and, moreover, with a seemingly obvious etymological link to annus.
In 1999, the Fountain of Anna Perenna, dating back to 4th century BC, was excavated in Rome, thereby consolidating the evidence that Anna could be Roman in origin. Although note that it has been suggested that Anna Perenna could even have been an Etruscan mother goddess originally and "her relationship with Aeneas was developed to strengthen her association with Rome." Nevertheless, it still stands that Anna does not have to be just the Latin for Hannah.