To add to the reply above, I think it is important to note that Harper Lee's mother's maiden name was Finch. Lee used all of her mother's name in her book. To me, this shows that her mother was very important to her. Yet the character Scout our narrator lost her mother at a young age.

With the finch being an innocent songbird, I often wonder if whatever struggles her own real mother had in life she tried to symbolize in her book by allowing her mother to die young. I think she certainly created her own family (Jem, Scout and Atticus) to be an innocent crew. They are disconnected as characters from the racial tension that occurs throughout the book and are named Finches.

Both are songbirds who belong to other songbird families.  Both make beautiful music and do no harm to other birds or farmers' crops.  These are symbolic of the Finch family, Dolphus, Tom, and Boo.  (See below)

They are contrasted by the bluejays, who are considered pests.  They do not make music; instead, they harm farmers' crops.  They travel alone.  This is symbolic of Bob Ewell in the novel.

From my New Oxford American Dictionary:

A finch is "a seed-eating songbird that typically has a stout bill and colorful plumage."  The true finches belong to the family Fringillidae (the finch family), which includes chaffinches, canaries, linnets, crossbills, etc. Many other finches belong to the bunting, waxbill, or sparrow families.

A mockingbird is "a long-tailed thrushlike songbird with grayish plumage, found mainly in tropical America and noted for its mimicry of the calls and songs of other birds."  The mockingbird family also includes the catbirds, thrashers, and tremblers.