The popular answer is that Ouija joins the French oui and the German ja, both meaning "yes." It is neat, and it is almost certainly wrong.
Etymonline notes that this explanation was already being offered in 1891, within a year of the word's first appearance. The Smithsonian says flatly that it is not true. Atlas Obscura calls it only the supposed origin and does not endorse it.
The name itself comes from Baltimore in 1890. According to Atlas Obscura, on April 25, 1890, the investor Charles Kennard, the attorney Elijah Bond and Bond's sister-in-law Helen Peters sat down with the board. The historian Robert Murch drew on a 1919 newspaper account by Kennard. In it they asked the board what it wanted to be called.
It spelled out O-U-I-J-A. When they asked what the word meant, the board answered "good luck." In other words, the name arrived with its own meaning attached, by the makers' account.
Peters then showed a locket she was wearing. It held a picture of a woman and the name "Ouija" beside it. Kennard took that as the board's answer.
Murch thinks the woman was Ouida, the pen name of the novelist Maria Louise Ramé, and that Kennard misread her signature. Etymonline raises the same possibility: that Ouida influenced the word. The Smithsonian says Peters admired the author. This is a theory, not a proven fact, but it explains the odd spelling better than French and German do.
Etymonline lists Ouija as a trademark name of the Kennard Novelty Company. Bond's patent for the board was granted on February 10, 1891. Wikipedia says the local patent office had first refused it. Bond and Peters, later known as Helen Peters Nosworthy, went to Washington, where a demonstration for the chief patent officer helped win approval.
Today the name belongs to Hasbro, which inherited it from Parker Brothers.
So the honest answer is that nobody can prove where the word came from. What survives is a séance story, a locket, a popular novelist, and a folk etymology that arrived almost at once.