One idea could be that a member of the bird family, like a hawk, has special powers and gets recruited by the Avengers. Maybe it can communicate with other birds and use them to gather intelligence.
Perhaps in the fanfiction, the bird family is from another dimension and the Avengers have to figure out how to work with them. For example, a family of phoenix - like birds that can control fire and they join forces with the Avengers to fight a common enemy. They could bring unique combat styles and powers that the Avengers haven't seen before.
Another interesting concept could be that a bird family is genetically modified by some evil force. The Avengers then have to save them and in the process, the birds develop a bond with the Avengers. Some of the birds could gain super strength or the ability to fly at supersonic speeds, adding new elements to the Avengers' team dynamics.