The Wikipedia article about the Iliad says that this work was first printed in Florence in 1488/89. The Wikipedia article about the Odyssey does not mention when this epic was first printed, but it appears to have been part of the same "editio princeps" as the Iliad: Homerus / Τὰ σωζόμενα (Onassis Library). The description of this edition in the Onassis Library also says that,
The edition of the epic poems was prepared by Demetrios Chalkokondyles, funded by his pupil Bernardo Nerli.
The article Printing Greek in the 15th century by Eugenia Russell on the British Library's website adds that,
In Florence, Demetrius Chalkokondyles (1423–1511) and Demetrius Damilas produced the first printed edition of the works of Homer (1488–89), using typefaces that the latter had used before in Milan. The production included all that was then believed to be Homeric, i.e. the Homeric Hymns and Batrachomyomachia (The Battle of Frogs and Mice), a parody previously attributed to Lucian. The Chalkokondyles Homer is probably the most famous editio princeps (first edition) in the history of philology.
However, what manuscript of the Odyssey was that edition based on? Or did the editors consult more than one manuscript? Does that manuscript still exist and, if yes, where is it currently available?
For what it's worth, How old is the oldest known manuscript for Iliad and Odyssey? on Answers.com mentions several manuscripts of the Iliad but for the Odyssey merely adds, "The story for the Odyssey is similar, but there are fewer medieval manuscripts and fewer papyrus fragments."
The Homer Multitext Project lists several manuscripts of Homeric works they are editing, but at the time of writing these are all manuscripts of the Iliad, not the Odyssey.
One potential candidate as a source is the "Florentine Homer", a manuscript by Ioannes Rhosos completed in Florence in May 1466, i.e. just two decades before the printed edition by Demetrios Chalkokondyles.