Big Bad: From a meta perspective, she was created in order to give the film a traditional main villain.
Cruel and Unusual Death: All her organic components were burned off, with her screaming the whole time. Her severed robotic head and neck appeared to still be functional afterwards, so Picard snapped the neck to deactivate her.
Hive Queen: Though she never claims to be the Queen.
I Am the Noun: "I am the Borg", and later, "I am the Collective."
Prim and Proper Bun: It's not actual hair, but the mass of wires on the back of her head are arranged in a circular pattern and she certainly has the personality to match.
Remember the New Guy?: Captain Picard remembers her when she's first introduced. Could be justified by the fact he was a Borg for a while, but that doesn't explain why he doesn't mention this vital piece of information about a dangerous enemy onscreen, and Data seemingly knows nothing about her despite the fact that he presumably would have read any report Picard made after the incident. They attempt to Handwave this by implying that Picard had forgotten about her until they're reintroduced in the film.
We Can Rule Together: Tries this on Data. It fails, though he seriously considered her proposal for 0.68 seconds. It's also implied that this was her aim with Picard when he was Locutus.
All There in the Manual: The novelization of First Contact reveals he actually suffers from Bipolar Disorder, and has a cranial implant designed to treat the disease. However, after the near societal collapse following WW 3, he no longer has the resources to maintain the implant. His alcoholism thus a desperate attempt at self-medication. At the end of the story, Dr. Crusher provides him with a permanent fix, reasoning that there is nothing in the history books saying Cochrane was mentally ill, and that an unmedicated bipolar diplomat dealing with famously stoic visitors could be disastrous.
The Alcoholic: He pilots the first warp-capable ship while hungover.
Broken Ace: He's an alcoholic misanthrope, dealing with the trauma of basically seeing his world end, who happens to be one of the best minds on the planet. He's also an Ace Pilot, capable of launching an untested prototype ship that can clear the earth's atmosphere, fly at warp, and then land, all hungover.
Surprisingly enough, this is mostly averted. While the Enterprise crew does look up to him, they also remember that he was a flawed human living in mankind's darkest period of history. So, while they do get annoyed with his Refused the Call attitude, they don't see him any differently then they did before.
Bunny-Ears Lawyer: Cochrane is one of the brightest minds of his time. He's also a heavy drinker who thinks first contact is best celebrated with whiskey and dancing. He also basically leads humanity out of the post-war Dark Ages.
Cassandra Truth: Enterprise would reveal that he attempted to tell people about what really happened during First Contact, but he was ignored and later recanted his claims. The crew of the Enterprise-E took great care to make sure had no hard evidence of his claims.
Crazy Awesome: He blasts music while launching the Phoenix, then gets drunk with the aliens he makes contact with.
Heroic B.S.O.D.: Has a minor breakdown after being confronted with time traveling heroes from the future who claim he's destined to lead the human race into a new era of history and enlightenment. It results in him trying to run away. Manages to get over it with the help of Commander Riker.
I Choose to Stay: At the end of "Metamorphosis". Given his negative feelings about being admired in First Contact, Kirk's line that "there's a whole galaxy out there waiting to honor you" is now Hilarious in Hindsight.
Jerk with a Heart of Gold: He starts off as a morose drunk, but proves he really does have the best intentions, and later becomes a distinguished diplomat. He is later shown to have mellowed out considerably in his later years.
Ladykiller in Love: If you regard the ending of "Metamorphosis" through the filter of his characterization in First Contact, he's a womanizer finally settling down.
Screw Destiny: Tries to do so after Geordi describes how famous he is in the future. ("You told him about the statue?")
Took a Level in Badass: Went from being a misanthropic, alcoholic jerkass barely eking out an existence to one of the foremost leaders of the human race during the reconstruction after World War III and the early space exploration days.
Who Wants to Live Forever?: In "Metamorphosis," he's discovered as a 237-year-old man trapped on a planet where he's become a de-aged immortal. He says that "immortality consists largely of boredom."
Cochrane: This other guy you keep mentioning, this historical figure? I never met him. I can't imagine I ever will. Riker: Someone once said, "Don't try to be a great man; just be a man, and let history make its own judgments." Cochrane: That's rhetorical nonsense. Who said that? Riker:[grinning] You did, ten years from now.
Lily Sloane (Alfre Woodard)
The Ace: Downplayed compared to Cochrane, but she is his partner in the Phoenix Project, and was supposed to be the one to copilot the mission.
Action Survivor: She survies World War III, radiation poisoning, and a Borg invasion.
MacGyvering: She and Cochrane were able to convert a nuclear missile into a fully funcitonal, warp-capable spacecraft, despite living in essentially the post-World War III dark ages. She even mentions having built the cockpit using titanium she scrounged from debris.
Ramming Always Works: Subverted. Her tactical collision against the Scimitar in Nemesis only takes out its shielding and cloaking device, but the Scimitar's main weapon remains intact and Shinzon intends to vaporize Earth once he destroys the Enterprise. Data saves her, at the cost of his own life.