The Longbottom Family
"I believe Neville visits [his parents], with his grandmother, during the holidays. They do not recognize him..."
-- Albus Dumbledore (GF30)
"I expect my Gran'd want me to try [to win the Triwizard Tournament] ... She's always going on about how I should be upholding the family honor."
-- Neville Longbottom (GF12)
Possible name meaning: Longbottom is a comical name, but could also indicate someone with "bottom," an old word for staying power. In Tolkien's Lord of the Rings cycle, 'Longbottom leaf' is a type of fine tobacco.
Frank and Alice Longbottom
Aurors, popular in the Wizarding World, who fought valiantly against Voldemort and his supporters during the 1970s. Frank and Alice were captured by Death Eaters shortly after the fall of Voldemort (late 1981). They were subjected to the Cruciatus Curse in an attempt to force them to tell where the defeated Dark Lord had fled to. The torture drove them insane and are now at St. Mungo's Hospital for Magical Maladies and Injuries in Ward 49. They do not even recognize family members. (GF30, OP23). Frank and Alice were members of the original Order of the Phoenix (OP, PC/JKR1).
Alice is a round-faced woman; her son Neville looks a lot like her, though by the mid-1990s her hair is wispy and gray. She does realize that Neville is someone she likes and gives him Drooble's Best Blowing Gum wrappers for gifts. Gran, Frank's mother, thinks of them as rubbish, but we get the feeling that Neville would save every single one if he could (OP23).
Augusta "Gran" Longbottom
Neville's grandmother who raised him in the absence of his parents. She is a forceful, strong woman and a powerful witch who wears green robes, a fox-fur scarf, a distinctive hat topped with a stuffed vulture, and carries a large red handbag. Neville loves her, but he's a bit scared of her as well. She is Frank's mother and she treats both Neville and his parents very unsentimentally. When Alice gives Neville a bubblegum wrapper, Gran dismisses it and tells Neville to throw it away (Neville doesn't listen) (OP23).
Neville came into his own after joining the D.A., and Augusta was quoted in the Daily Prophet saying very positive things about him. In the spring of 1998, Death Eaters sent Dawlish to try to take her into custody. It didn’t go so well for Dawlish. As a result, a few weeks later Dawlish was still in the hospital (DH29), while Gran fought in the Battle of Hogwarts (DH31).
On Rowling's Black Family Tree, a Harfang Longbottom married Callidora Black, daughter of Arcturus Black. No dates for Harfang, but Callidora was born in 1915 and appears to still be alive (no death date) (BFT).
Possible name origin: In French, an 'Harfang des neiges' is a snowy owl. In the Chronicles of Narnia, the House of Harfang was the large (even by giant standards) castle of a clan of Northern Giants.
Under interrogation, Neville tells Umbridge that he can see Thestrals because he saw his Grandad die (OP21). This may have been Augusta's husband, or his grandad on his mother's side.
Great Uncle Algie
Algie tried to trick Neville into showing some magical ability as a child. Pushed him off the end of Blackpool Pier once and nearly drowned him. Algie held Neville out of an upstairs window when he was eight and accidentally dropped him, only to see the boy bounce all the way down the garden and into the road; this was cause for great rejoicing in the Longbottom family. Bought Trevor the Toad for Neville (PS7) and brought a Mimbulus mimbletonia (OP10) for Neville.