This strikes me as quite a challenging question. From the outset of Homer's Odyssey, the poet characterizes Odysseus as a hero whose intellectual prowess surpasses his physical prowess. He is a man who has endured much on both land and sea.
If he does have intellectual weaknesses, then I suppose they show up in two possible areas. I'm not sure if not being able to stay awake would be considered an intellectual weakness, but Odysseus does fall asleep at two critical points during the epic. First, in Odyssey 10, after acquiring the bag of winds from Aeolus, Odysseus falls asleep as he and his crew are practically within sight of his native land of Ithaca. With Odysseus asleep, his men become curious about the bag, which releases the winds and blows them far away from home again. Second, in Odyssey 12, while on the island of Thrinacia, Odysseus again falls asleep, which allows his hungry men the time they need to kill the cattle of Helios, an action that causes that god to become angry arrange for the destruction of the rest of Odysseus' men and ships.
Another possible intellectual weakness is that sometimes Odysseus becomes too curious. This manifests itself in the land of the Cyclopes (Odyssey 9) when Odysseus wants to wait for the Cyclops to return to his cave, despite the fact that "my men begged me to take some cheeses and go". Odysseus, however, "would not listen."
Later, in the same book, Odysseus himself makes a critical mistake when he reveals his real name to the Cyclops. This revelation allows the Cyclops to identify Odysseus to his father, Poseidon, who ends up striking Odysseus' raft after he leaves Calypso's island in Odyssey 5.
So, if Odysseus does have intellectual weaknesses, I would suggest that they are an insatiable curiousity that sometimes causes him to endanger himself and his men. He also exhibits a mental lapse, when he reveals his name to the Cyclops. Finally, if falling asleep at critical moments can be considered an intellectual weakness, then I suppose Odysseus is guilty of this.