Now that's not even remotely true. Nowhere was it ever indicated in the books or the movies that Ginny was the favorite Weasley child.
How often was Mrs. Weasley doting over Percy's good behavior and status as both a Prefect and Head Boy? And the same when Ron was made a Prefect in Order of the Phoenix?
Or in Goblet of Fire, when Mrs. Weasley and the twins had fought before they left for the World Cup and weren't speaking. When they came back after the Death Eater attack, she was crying how upset she was that they had left angry and how terrible she felt if those had been the final words they'd had with each other?
Remember in Order of the Phoenix, when Mrs. Weasley found a Boggart in one of the rooms in Grimmauld Place and it kept turning into the things she feared most? It turned into the dead bodies of every member of her family, and no matter how hard she tried, she couldn't make it stop. She was inconsolable at the thought of losing any of them. Someone else had to use the Riddikulus charm to put it out, because she was crying too much to do it herself.
Or in Half-Blood Prince, how she at first thought how Fleur wasn't good enough for Bill, until that moment after Bill was bitten by Greyback?
Why would she take Harry in if she already had a "favorite" in Ginny and didn't care as much for the others? She was the one who came as the family member (along with Bill) for the last event, and held him in her arms when he started to cry in the hospital wing over what happened to Cedric. Harry realized at the time that it was the first time he'd ever known as being held, "as if by a mother."
And on Harry's 17th birthday, she had given him a watch from one of their own family, which was a tradition Harry wouldn't have been able to have since his family was gone.
If Ginny was doted on differently, it was because she was the youngest. And she was possessed by Voldemort in her very first year at Hogwarts, which would certainly make them more watchful over Ginny since she'd been attacked. Although, it's worth nothing that the Weasley came to Hogwarts when Ginny was kidnapped and after Ron was poisoned in Half-Blood Prince.
As for the movie-universe, Bellatrix was one of the Death Eaters who had burned down their house and could have killed her entire family. That's enough for a few naughty names directed her way as well.
She didn't go after Bellatrix just because of her daughter. She did it for everyone.
The "daughter" part was because Bellatrix had just tried to kill her daughter. A moment later (in the book), she had said "You will never touch our children again!"
She needed to end it, once and for all.