The unnamed narrator is just one of the dinner guests, so why does he get his own character analysis? Like the other guests, he's mostly left undescribed, and we don't even know what he does for a living. He says he has an appointment with a publisher (12.28), so he might be a writer. He also goes to the Linnaean Society (2.1), though, so he might be a scientist – we just can't know for sure.
One thing that sets the narrator apart from the other guests is that he sort of believes the Time Traveller's story. Everyone else makes jokes, thinking the Time Traveller is pranking them or is simply stressed out. But the narrator is at least willing to entertain the idea that time travel is possible. He's like the perfect science fiction reader – somebody who's ready to suspend his disbelief for a moment and consider something out of the ordinary. In this way, the narrator seems more like the Time Traveller than the other guests. This might be why the narrator enjoys the Time Traveller's company so much; as he notes, he "was one of the Time Traveller's most constant guests" (2.2).
On the other hand, as the narrator notes in the Epilogue, while the Time Traveller is a pessimist who thinks human progress is an illusion, the narrator is more of an optimist, who thinks things have to get better. (For more about those "things," check the "Setting" page.) The narrator gets the final word in the novel, and it's a mixed but optimistic message: "even when mind and strength had gone, gratitude and a mutual tenderness still lived on in the heart of man" (Epilogue.1). As we said, that's not a super-happy ending, but it's probably a happier one than the Time Traveller would have left us with.
There's a possibility that the unnamed narrator's optimism is related to the Time Traveller's disappearance. Check out Chapter 12, when the narrator looks in on the Time Machine all alone:
I stared for a minute at the Time Machine and put out my hand and touched the lever. At that the squat substantial-looking mass swayed like a bough shaken by the wind. Its instability startled me extremely, and I had a queer reminiscence of the childish days when I used to be forbidden to meddle. (12.25)
Now, this is just a possibility, but do you think the narrator may have tampered with the Time Machine? Maybe he didn't mean to (or maybe he did) but when he touches the lever, what if he messed something up, and that's why he has "a queer reminiscence of the childish days when [he] used to be forbidden to meddle."
In Chapter 12, when the Time Traveller returns from the future, he seems "to see Hillyer for a moment; but he passed like a flash" (12.2). This is the only mention of a character named Hillyer, so the only thing we know about him is that he enters the Time Traveller's lab by himself. Hillyer could be the name of a servant, but it's possible that it's the name of the unnamed narrator, who enters the lab by himself twice (12.25 and 12.29).
If that's the case, why doesn't the Time Traveller see Hillyer twice? Well, maybe the Time Machine is moving so fast that it looks like he only comes in once. Or maybe Wells made a mistake. (After all, Wells is writing one of the first books about time travel, so we'll cut him some slack.) Or maybe Hillyer is someone else entirely. We think Hillyer is the narrator, but we're open to other possibilities.