Obi Wan is a wise character for a variety of reasons. They all could be summarized by saying that he is the epitome of what a Jedi is. The list of Jedi that I feel epitomize its tenets, and are therefore wise, is fairly short. Mace Windu, Obi Wan, Qui Gon Jinn, and Yoda are about the only Jedi that I feel are exceptionally wise characters.
I believe that Obi Wan is a wise character because he is patient. He doesn't rush into things, because he thinks through his actions before acting. That's what sets him so apart from Anikin Skywalker in Episodes 1-3. Anikin is impetuous to a fault, and it almost never ends up benefiting him. Obi Wan, on the other hand, thoughtfully weighs all of his options and chooses the best choice. That makes him wise. His decisions aren't always well liked, but it is clear that they are always the right choice.
Obi Wan is also a wise character because he knows when to ask for help. After Vader "kills" him, Obi Wan still has some kind of Force connection with Luke. Obi Wan could continue to train Luke through that connection, but instead Obi Wan sends him to Yoda. He knows that Yoda can help him more. Obi Wan isn't so full of pride that he can't seek help from others. That makes him wise.