So what makes River so special? After much consideration, I think I’ve figured out what it is. River Song exists both as a superhuman foil to the Doctor, and someone we can relate to as a person who loves and admires him. As both, how can we resist her?
How River is More Than Human
The Doctor is an amazing character, I will not start in on how spectacular he is. He regularly has a companion with him which offers the audience someone to compare him to. In contrast, he is the ever energetic adventurer and savior. We see different sides of him as he is paired with various companions, but the Doctor is always the savior, the hero in the relationship. He opens up worlds for his companions, and expands the universe.
River is so different from anyone else in the series. As far as we’ve been allowed to see, River doesn’t need the Doctor to expand her universe. Her universe is already vast, covering a great deal of time and space. She has her own adventures and is capable of travelling and existing in her own timeline, with and without the Doctor. In the new series, there are no Time Lords other than The Doctor. It wasn’t until a ways into viewing that Jana & I discovered that this wasn’t always the case. Russell T. Davies made a decision that the race would not survive the Time War, and it’s universally been embraced. In Classic Who there were many other Time Lords. They were capable of time travel, had working Tardises, and had the same vast knowledge base and abilities as the Doctor. They did not always see the universe in the same way as him, but they offered the viewer a look at a universe were the Doctor was not a single, ultimate being. It’s what makes the villain of the Master so intriguing in the new series- here’s an antagonist who can hold his own against the Doctor. They begin on equal footing. We get a glimpse of the Time Lords at the end of season four, but it's just for a moment.
With the appearance of River Song, we have a character who operates on her own agenda, with her own abilities, and doesn’t need the Doctor to physically survive the universe. In this sense, she is completely unique in the New Series. Captain Jack was around for a while, but he (very unfortunately) hasn't been seen in the company of the Doctor in ages. We need this character, River, to remind us that the Doctor is not the beginning and the end of intelligent, powerful beings. He is able to ask questions of her and get them answered. Their relationship is back and forth with knowledge and information in a way he’s never had with his earthly, contemporary companions. She understands situations that they are in without needing a ton of background, and therefore he is able to bounce ideas off of her. He’s able to question options and have someone make suggestions about what to do. The Doctor has less pressure on him with River there, and while he ultimately feels responsible for her, he isn’t really. She takes care of him just as much. The first episode she meets him, she does the saving (granted, she also tried to kill him...). The fact that he doesn't have to be the savior gives him the rare chance to be vulnerable around someone. This makes their relationship special.
What makes River especially amazing is the fact that she can run around the universe as nearly an equal match to the Doctor, but she is still a character that we are able to relate to. She adores the Doctor, pretty much unconditionally. What true Whovian doesn’t unconditionally love the Doctor, even when he screws things up? She is always excited to see him, and she will do anything for him. The awesome part is that she isn’t afraid to argue with him. She is knowledgeable enough to do it, so watching her and the Doctor together is more like watching a real couple than anything we’ve seen before.
Another detail is that she isn’t eternal. River is human (well, as much as a Child of the Tardis can be), and she knows that her time with the Doctor will someday come to an end. Granted, with her crazy timeline we may be able to see her for ages, but she still isn’t going to live for thousands of years like the Doctor. And everyone, including the Doctor, knows it.
River, unlike the Doctor and very like real people, can regularly be selfish. With her “trained as a psychopath” childhood, if she wants to kill someone/something, she does. If the Doctor is going to die, he might as well take all of time and space with him. She’ll deface ancient or futuristic property alike, and she is unconcerned. She’s manipulative to the core (see regular habit of lying and her steady supply of hallucinogenic lipstick), and River is not all that concerned with applying her moral compass.
All these aspects together are what make River a spectacular character. She’s strong, self-assured, and capable like the Doctor, but she’s mortal and has human flaws. River, of all the characters I’ve ever seen on Doctor Who, is the character I most want to be.