Arthur Holmwood gets the girl... and then kind of fades out of the picture. So why is he important? Because he's connected.
Arthur Holmwood goes by so many different names in this novel that we'd better pause and clarify here. His father is Lord Godalming, a wealthy aristocrat. While his father is still alive, at the beginning of the novel, Arthur just goes by his first and last names, Arthur Holmwood (although, as the son of a lord, he can sign his name "the Honorable Arthur Holmwood").
After his father dies, though, Arthur inherits his father's wealth and his title, so he becomes "Lord Godalming." But even after Arthur's name officially changes, the rest of the characters still call him "Arthur." Dr. Seward and Quincey Morris have been friends with him for ages, so it makes sense that they'd call him Arthur. But Van Helsing and the Harkers have just met him—they occasionally refer to him by his official title, but he prefers to be called "Arthur." Why is this? What does this say about him? Maybe that he's more down-to-earth than your average nobleman, or maybe that he prefers the individuality of his first name, Arthur, to the label that he's inherited.
Now that we have his name straight, what can we say about Arthur's character? Well, not nearly as much as about some of the others—after all, Arthur doesn't contribute any journal entries to the novel, and only a couple of letters, so we rarely get to see things from his point of view. That's why readers tend to feel more sympathy for Dr. Seward than they do for Arthur when Lucy dies, even though Arthur is actually Lucy's fiancé.
But we do see Arthur break down and cry in one memorable scene—after Lucy has died, Arthur has no one to talk to about it. And when he meets Mina for the first time, she offers her sympathy, and he simply breaks down and cries like a baby. He doesn't apologize for his "weakness" (real men weren't supposed to cry in Victorian England); instead, he just offers to be like a "brother" to Mina for the rest of their lives (17.60).
So even though we don't see much from Arthur's perspective, we do have glimpses of his personality and character that make him more sympathetic and interesting. Scenes like the aforementioned make Arthur seem human, instead of simply an empty-headed aristocrat.