PERRY, Ga. — As a little girl, Michelle Nunn clutched her father’s hand as they strolled the pecan groves of the family’s 2,400-acre farm, a moment captured in a black-and-white photograph that Sam Nunn used to help win election to the Senate. It hangs in a Sam Nunn exhibit in this small city south of Macon, where the main thoroughfare is Sam Nunn Boulevard.

It all speaks to another era. Mr. Nunn, a Democrat, last appeared on a Georgia ballot in 1990, and Georgians have since fled his party. Now his daughter, a political novice who spent 25 years running an Atlanta-based nonprofit volunteer network, is hoping to capitalize on the family name to reverse Democrats’ long decline in the Deep South.

With $3.3 million in the bank, Ms. Nunn is the party’s best hope in an open race to replace Senator Saxby Chambliss, a Republican, who is retiring.

Her campaign will test whether the rapidly changing demographics of Georgia — where state elections data show that the white vote dropped to 61 percent of the total in 2012 from 75 percent in 2000 — have shifted enough to return a Democrat to Washington. And it will reveal how much legacy still matters in politics.