One common theme is their relationship. Writers often explore how they support each other through the traumas they've experienced in the Hunger Games world. For example, Finnick's protection of Annie.
One common theme is his trauma. Given his experiences in the Hunger Games and his exploitation, many fanfics explore how he copes with that. Another theme is love. Finnick's relationships, whether it's his love for Annie or his friendships, are often central in fanfics. And there's also the theme of resistance. As a strong character, he is often depicted as part of the resistance against the Capitol in fanfics.
One common theme is trauma recovery. Given Annie's experiences in the Hunger Games, many fanfics explore how she deals with the mental and emotional scars. For example, in some stories, she has nightmares and has to learn to trust people again.
Fanfiction.net is also a great place to look. A lot of Hunger Games fanfic writers post their works there. Just be aware that the quality can vary a great deal. You might need to sift through some stories to find the really good ones about Annie and Finnick.
A lot of fanfics deal with his charm and charisma. Writers often expand on how he uses these traits to survive in the Capitol's cut - throat world. There are also themes related to his internal struggles. For example, how he copes with the loss of his innocence due to the Games and the Capitol's exploitation of him. And of course, love stories involving Finnick are quite common, whether it's a new relationship or the development of his relationship with Annie.
One common theme is the struggle between his human side and his vampire instincts. Finnick often has to fight against his newfound bloodlust while still holding onto his humanity.
One common theme is protection. Finnick is often depicted as very protective of his sister, given the dangerous world they live in. For example, in many stories, he shields her from the Capitol's cruel games and demands.
One common theme is trust. Given Finnick's past and the dangerous world they live in, building trust between Finnick and the reader is often a central part of the stories. Also, sacrifice is a theme. Finnick might be willing to sacrifice himself for the reader, or vice versa. And of course, the theme of hope, that their relationship can bring hope in a dark world like Panem.
One common theme is trauma. Given what Annie has been through in the Hunger Games, many fanfics explore how she copes with the mental and emotional scars. Another theme is love, especially her relationship with Finnick.
A third common theme is her connection to the water. Since she's from a district known for fishing, many fanfictions use her love for the water as a symbol of her strength and also as a way to heal from her traumas. In some fics, she has a spiritual connection to the water that helps her move forward in life.
Romance can also be a theme, not just in the sense of a romantic relationship between Peeta and Finnick, but also how their relationships with other characters are affected by their own experiences. Some fan fictions might explore Peeta's love for Katniss and how Finnick's own relationships are intertwined with the events, creating complex emotional webs that add depth to the story.