In the fanfiction, Percy might use his powers over water to somehow rejuvenate Artemis. Maybe he discovers a magical spring that has the power to stop her from fading.
One consequence could be that Artemis becomes indebted to Percy. This might lead to her providing him with special favors or assistance in his future adventures. For example, she could offer her hunters to help Percy in a battle.
In the fanfiction, Artemis might use her divine powers. Maybe she creates a distraction so that Percy can escape from Gabe. For example, she could make a sudden bright light or a loud noise that confuses Gabe, giving Percy the chance to run away.
Maybe Artemis sees Percy's potential. Percy has shown great courage and skill in other situations, and Artemis might recognize that he could be an important asset in the future battles against the forces of evil. So, she decides to save him from Gabe, who is a nuisance and a threat to Percy.
The Hunters of Artemis are very likely to show up. They are Artemis's loyal followers and would surely be involved in the search for their mistress. Annabeth could also be there. She's Percy's friend and has great wisdom, which could be useful in devising a plan to save Artemis. Grover might be another character. His ability to sense things could help in tracking Artemis or finding clues about her whereabouts.
One possible plot could be that Artemis is captured by some powerful ancient enemy, like a primordial being. Percy, with his skills and courage, embarks on a quest to find her. He might face various challenges and monsters along the way. Maybe he has to solve some riddles related to the moon, as it's Artemis's domain. Finally, he locates Artemis and uses his water powers in a creative way to defeat the captor and free her.
I haven't actually read this specific fanfiction, so I can't summarize the exact plot. But generally, it might involve Percy being in a very dark and suicidal state, and Artemis somehow intervenes. Maybe she uses her powers or her wisdom to show Percy that there are reasons to keep living, like the importance of his friends, his potential to do great things in the future, or the value of his own life in the grand scheme of their fictional world.
It could be that Percy has been ostracized or misunderstood by the people he cares about. If his actions were misinterpreted and he was made to feel like an outcast, that might have led to his suicidal thoughts. Also, he might be suffering from some kind of prophecy or fate that seems inescapable, and he thinks death is the only way out.
Annabeth is quite popular. Since she has a relationship with Percy in the original series, in fanfiction related to Percy and Artemis, she can be a source of conflict or support. For example, she might be jealous of the time Percy spends with Artemis. Another popular character is Thalia. Her connection to Artemis as a member of the Hunters makes her an important figure. She can add more depth to the stories, especially when it comes to the dynamics between the Hunters and Percy. And then there's Grover. His loyalty to Percy and his unique abilities as a satyr can bring a different flavor to the fanfiction. He might be the one who tries to mediate between Percy and those who are skeptical of his association with Artemis.
There are many fanfic platforms where you can find 'Percy Saves Artemis' fanfics. For example, FanFiction.net. You can simply search for the specific tag and browse through the available stories. Some might be action - packed with Percy using his demigod powers in creative ways to rescue Artemis.