Atticus saves Miss Maudie's oak rocking chair because, he, like Scout, understands that it is "what she value(s) most" (Chapter 8).
Miss Maudie is a genuine, kind, and sensible woman. A professed Baptist, she lives the true spirit of religion, as opposed to "foot-washing Baptists" who "believe(s) anything that's pleasure is a sin", and characters like Miss Stephanie, who hypocritically exhibit their righteousness while harboring meanness and prejudice. Miss Maudie "love(s) everything that (grows) in God's earth", and likes nothing better than to sit in the evening twilight in her old oak rocking chair on the porch, enjoying "God's outdoors" and fellowshipping with those who come her way. Miss Maudie treats the Finch children with dignity and respect, and Scout remembers many summer evenings sitting in hospitable companionship "with Miss Maudie Atkinson on her front porch" (Chapter 5).
The rocking chair can be considered a symbol of sorts, representative of Miss Maudie's love of nature and respect for community and friendship. The rocking chair may not be worth a lot from a material standpoint, but from a sentimental perspective, it is priceless.
Miss Maudie is a kind person who looks above race. That chair was very important to her and Atticus went in there and saved it. Miss Maudie has always treated the Finches kindly and with respect and while the chair does not look like much, it means a lot to her in terms of compassion and friendship.