In The Odyssey, how does the character of Odysseus change from the beginning of his quest until the end?
I'm honestly not sure that Odysseus does change very much during his travels. It's true that, in the end, he does display prudence and patience while awaiting his opportunity to avenge the wrongs done to his home by the suitors. However, Odysseus has shown prudence and patience before, when necessary; these aren't really new qualities that he's developed. For example, when he's in Polyphemus's cave, he prudently realizes that, much as he wants to kill the monster, if he does so, he and his men will be trapped in the cave. He needed to discover a way to maim the monster without killing him so that the Cyclops could still move the boulder away from the door. Odysseus patiently awaits his chance to blind the monster, and he is successful in his plan. He also knows that he and his men cannot rush out the door when it is open, because the Cyclops will grab them. He patiently awaits the opportunity to leave, prudently tying himself and his men underneath Polyphemus's sheep rather than on top, where the monster's hand might feel them.
Likewise, prudence and patience are necessary when dealing with the suitors. If he rushes in, holding his sword aloft, the suitors could defeat Odysseus as a result of their greater numbers. Again, he must exercise these qualities—qualities he's had all along—because, if he doesn't, he could die. He is brash, and he does have bravado, and he hasn't always been prudent in all situations, but it is a characteristic he's possessed for a long time. Then, the manner in which he and Telemachus decimate the suitors' numbers surely gratifies that bravado, which he's always possessed as well.
To answer this question we need to understand that one of the key desires of Odysseus as a character, which is a desire that he shares with all Homeric heroes, is the desire to gain glory through his exploits and deeds. At various points through the text, we see the way in which Odysseus rather arrogantly ignores the advice of different characters or does what he thinks is best without thinking about the consequences of his actions and what might happen to both himself, and to his men, as a result. The best example of this is when, at the start of his wanderings, Odysseus was so focussed on gaining glory that he deliberately revealed who he was to the Cyclops, and as a result brought down calamity on both himself and his men because of Poseidon's special relationship to the Cyclops.
However, by the end of the story, when he reaches Ithaca, the way in which he deliberately bides his time and disguises himself as a beggar instead of rushing into his home and declaring who he was before the group of extremely hostile suitors shows how he has developed and changed. He is not the same glory-hungry individual who often committed rash and rather vacuous actions in order to gain kudos. Instead, he tempers this aspect of his character with the need for patience, and remains incognito until he has established his position and what men are loyal to him and he is ready to strike. The text therefore indicates that Odysseus as a character very definitely learns from his mistakes and develops during the course of the epic.