Charlie could use magic to simply apparate Harry away from the Dursleys' house. It's a quick and efficient way to get Harry out of that horrible situation.
Perhaps Charlie disguises himself as someone the Dursleys trust, like a social worker or something. Then he can easily take Harry with him under the pretense of some official business.
Charlie might bring some of his dragon - taming skills into play. He could create a distraction using his knowledge of magical creatures. For example, he could make it seem like there's a dragon on the loose near the Dursleys' place. While they're all panicking, he grabs Harry and gets away.
Charlie might introduce Harry to the dragon world. Since Charlie works with dragons, this could be an exciting new adventure for Harry. It would also give him a new perspective on the magical world and something to focus on other than his difficult past with the Dursleys.
Edward could use his mind - reading ability. He reads Charlie's intentions and plans his moves accordingly. If Charlie is about to make a decision that Edward believes will be bad for Bella, he can intervene at the right time. He might also be able to manipulate the situation subtly so that Charlie changes his mind without realizing it.
In some fanfictions, Fleur might use her Veela charm to intimidate the Dursleys into letting Harry go. The Dursleys, being easily cowed by the extraordinary, would be no match for Fleur's alluring yet powerful presence.
Well, in some fanfictions, Charlie might use his dragon - handling skills to protect Harry. For example, he could train dragons to guard Harry or use his knowledge of magical creatures to fend off threats that are related to dark magic and its associated beasts.
Charlie might be a character who has known Hermione for a long time, maybe through some other connections in the wizarding world. He sees Hermione as someone in need of protection from Ron, whether Ron is acting out of his own accord or is being influenced by external forces. In relation to Ron, he is the one who stands against Ron's actions towards Hermione.
Perhaps Snape uses magic to create a distraction at the Dursleys' place. He could make strange noises or apparitions that scare the Dursleys, and while they are in a panic, he grabs Harry and disapparates. This would be a quick and efficient way for him to rescue Harry.
Maybe Percy uses his magic skills to create a distraction. He could conjure up some strange creatures outside the Dursleys' house. The Dursleys, being afraid of all things magical, would be distracted long enough for Percy to sneak in and get Harry out.
Well, he could use his magic skills secretly. Maybe he casts a spell on the Dursleys to make them docile or distracted long enough to get Harry out. For example, he might use a Confundus Charm on them so they don't notice Harry leaving.
In the fanfic, Snape might use his knowledge of the Dursleys' house layout. He may enter the house unnoticed, find Harry, and then use a Portkey to quickly get them both out of there. He could also have some sort of charm or potion that he uses to counter any magic the Dursleys might have used to keep Harry there, like a spell - breaking potion. And he may have planned the whole rescue very carefully, choosing a time when the Dursleys are least likely to notice, like in the middle of the night.
Well, in some fanfic scenarios, Voldemort might have made a deal with someone who has access to the Dursleys' house. For example, he could have bribed or threatened a neighbor or someone who works in the area. Then, when the time is right, he enters the house under the guise of this person's help and rescues Harry. It could also be that Voldemort has some sort of inside knowledge about the protective spells around the Dursleys' place. He finds a loophole in those spells and slips in to save Harry.