Not much is known about her, except that she's a deceased relative of the Loud family. Lucy still treats her as a living person, which indicates that, (assuming Lucy and her siblings knew her when she was still alive) she and Lucy were very close.
In "Snow Bored", Lucy fruitlessly calls her great-grandmother to save her and Lynn from Lisa's incoming big snowball.
In "Spell It Out", it is revealed that she once owned a spell book, meaning that just like Lucy, she was also interested in occultism.
When Harriet was a child, she bears a striking resemblance to Lucy, with a few exceptions: Harriet was seen wearing pearl earrings, a pearl necklace, a black dress, with a brown waistband and brown cuffs, and a brown 1940s hat, as opposed to Lucy's outfit.
So far, she has only appeared in a photo.
She is the first mentioned deceased character belonging to the Loud Family.
"Spell It Out" reveals that she resembled Lucy at a young age. It seems (though she's deceased) she is closest to Lucy, as they look and act alike, and form a somewhat spiritual connection.