Governor Swann wants a lot of things for his daughter. He wants her to be strong and intelligent. He wants her to be well-connected and taken care of. He wants her to be healthy. But, above all, he wants her to be happy.
To these ends, Norrington seems like the answer. It's a smart match, with friends in high places. He can give her everything she needs and anything she asks for, and it's obvious he adores her. It's a suitable match.
But it's obvious someone else adores her as well. An orphan, a blacksmith, with no connections anywhere. Sure, he can put a roof over her head and food on the table, but the dresses will be simple and the jewelry minimal.
He's just not good enough for her.
But the truth is that no one ever will be. Norrington is appropriate, suitable at best. It's a smart match, as his peers have reminded him often. He can take care of her. Additionally, he's fond of her, so he'll be sure to do what he can to make her happy. Of course he's the most suitable husband for her.
Then Will Turner comes in, showing his hotheaded and passionate side, which mirrors Elizabeth's (when she has those moments). The commodore can't match her in that.
And Will Turner risks life, limb, and reputation to rescue her. He does whatever it takes, even outside of the law, to get her safely back home.
The governor remembers how Elizabeth's face used to light up whenever Will appeared. He remembered how happy she always was in his presence.
And he certainly loves her. He has proven. That is in his favor.
So the governor rethinks his position. If his daughter must have this one, perhaps Will Turner, orphan, blacksmith, pirate, could be a suitable match for her after all.