The mockingbird is cited as an example of a specific set of traits in the novel.
As Miss Maudie Atkinson explains, it would be thoughtlessly cruel to kill innocent creatures that "don't do one thing but make music for us to enjoy."
Mockingbirds are harmless and in need of protection though they do good things for the small world around them. Many characters from the novel end up fitting this description, from Boo Radley and Tom Robinson to Atticus and Scout Finch.
Choosing other animals that could fill the same symbolic place, you will want to choose those that possess the traits of harmlessness, a need of protection, and beauty. In your explanations, you will describe what beauty the animal brings into the world and how they are harmless and/or need protection.
Other song birds could easily be selected as stand-ins for the mockingbird. Peacocks or other animals with noticeable plumage or coats could work as well. Many woodland creatures might also fit.