It likely alludes to a Phoenix -- a mythical bird that never dies.
In the Arabian myth, It was said that only one phoenix existed at any one time, and it is very long-lived with a life span of 500 to 1,000 years. Before it's "death", the phoenix would build a nest, set it on fire, and be consumed in the flames. After three days is the birth -- or as some legends say a rebirth -- the phoenix rises from the ashes. At this point, the phoenix arose from the midst of the flames and is 'reborn'.
In the book, Harper Lee is saying "My feathers rose again" alluding to the Phoenix being "reborn". The character (Scout, I presume?) is being re-born, coming back to life, ready to tackle the next challenges.
Hope this helps.
Depending on the situation, if I had to guess it could either mean, he was so excited about something that 'his feathers rose again' or maybe he was so afraid of someone or something that his 'feathers rose again' as like, another way of saying 'the hair on the back of my neck stood up again'. You know what I mean? I've never read they book or heard the saying so, again if I had to guess, that's what I'd go with. =)
what chapter is it in? what stage of the book is it?