I have asked this question on the English Wikipedia Reference Desk a few months ago. This answer contains a copy of the answers volunteered there.
**Livius Andronicus** (c. 284 – c. 204 BC) was possibly the first who translated the Odyssey into Latin, but his translation has not survived. There have been many Latin translations of Homer over the centuries, but the **oldest surviving** seems to be by **Leontius Pilatus** (died 1366): see catalog entry of a modern publication of that translation. This answer contributed by [Lindert.]
Educated Romans read Homer in the original Greek, so I'm not sure there was great demand for a written literary translation into Latin. There was the Ilias Latina, which was kind of "downmarket" (not a full translation, and not very literary). This answer contributed by [AnonMoos.]
Attius Labeo was renowned for the badness of his translations of Homer, so there was clearly a market. Comment by [Paul Barlow.]