Aunt Drip is a deceased relative of The Addams Family.
She was married to Uncle Droop. A portrait of her hangs in the living room, above Uncle Droop's urn. Another portrait of her hangs behind two of her husband's paintings that hide the Addams family safe.
She presumably lived during the 19th century (her portrait shows that she's dressed in an 1850s-1860s style), with her husband who may have been in the Civil War.
The Addams Family original series episodes: