Well, in many such fanfictions, Gaara might use his sand powers. He could sense Naruto is in danger through his connection to the sand or his own instincts. Then, he'd quickly move in, using his sand to shield Naruto from any attacks by the Akatsuki members, like blocking their jutsu with his powerful sand barriers.
One reason is their friendship. Naruto was one of the first people to see Gaara as a person, not a monster. So, Gaara would feel indebted to Naruto and be willing to risk his life. Another reason could be for the sake of the ninja world. Naruto is often seen as a key figure in maintaining peace, and Gaara, as a responsible leader, would understand the importance of saving him.
Well, a plot twist might be that Naruto actually gets captured by the Akatsuki while trying to save Gaara. Then, Gaara, in his weakened state, has to find a way to save Naruto instead. This would completely change the dynamic of the story. Or, it could be that the Akatsuki's real target isn't Gaara at all, but Naruto, and they use Gaara as bait. So when Naruto arrives to save Gaara, he walks right into their trap, but then uses his wits and new jutsu to turn the tables on the Akatsuki.
In many Sasuke - Sakura - Gaara fanfictions, Sasuke saving Sakura from Gaara can be a really exciting plot. It often shows Sasuke's strength and his hidden feelings for Sakura. Maybe he uses his Sharingan to outwit Gaara's sand attacks and rescue Sakura just in time.
One great example could be a fanfiction where Naruto has to battle a powerful new enemy who has kidnapped Gaara. Naruto uses his new jutsu and his unwavering friendship to break through the enemy's defenses and save Gaara. Another might be a story set in an alternate universe where the circumstances of Gaara's capture are different, but Naruto still goes to great lengths, even sacrificing his own safety, to rescue Gaara.
Naruto's character could develop in terms of his strategic thinking. Since saving Gaara from the Akatsuki is no easy feat, he might have to come up with more complex battle plans. This would show his growth from a brash fighter to a more calculating ninja.
Their relationship would likely deepen with trust. Naruto would be extremely grateful to Gaara for saving him, and this would strengthen their bond. They might start sharing more about their past traumas and how they overcame them.
A common plot point is danger. Fem Gaara is often in some sort of life - threatening situation, like being attacked by enemies. Naruto then uses his skills, such as his Rasengan or his ability to make friends out of foes, to save her.