I think the characters I like the most are the ones that we get to know and get to see grow throughout the time that they appear on the show. As an example, I really like both versions of Pete Tyler, but I prefer the one from "Father's Day" because of his character development and growth; the Pete's World version started strong and stayed strong - he didn't have much to learn and didn't. (Other examples: Donna and Rory.) A few of the characters that I really like are Luke Rattigan, Blon Fel-Fotch Passameer Day Slitheen (who grows and develops but does not suddenly become good, staying true to herself), Quintus Caecilius, and especially Mickey Smith.
The one I love the most, though, is Harriet Jones. She appears first in "Aliens of London" as an MP from a nowhere area in England and while she's strong enough to try to bull her way through the PM's office to get what she needs, she knows she's a nobody and is appropriately deferential. When she first encounters the Slitheen, she begins to panic. Once the Doctor has established his war room, though, she takes charge, making decisions on behalf of the people.
She appears again once she's become the Prime Minister, demonstrating her devotion to her people ("Never off duty.") and taking charge of the Sycorax situation. She makes the decision to destroy the Sycorax ship, even though she obviously doesn't want to kill them, and opposed the Doctor, maintaining that what she did was right for her planet.
She appears again when the planet is under attack from the Daleks, showing that she's spent her retirement developing the network to contact all of the important people during crises. She continues to advocate the idea that the Earth must learn to defend itself, while also attempting to contact the Doctor because she knows the planet needs his help. And finally, she sacrifices herself for the rest of the team.
This is a brilliantly-designed, strong character, and she is my very favorite of all of the people the Doctor has encountered in the modern show.