At first, we know Sef only as Nan's "made-up friend" (1.7), but actually, Sef was a real person, May's best friend and protector at the orphanage where she spent her childhood. Sef was two years older than May, and even in her old age, May vividly recalls her "in her long, white nightdress sitting on the edge of May's bed, holding her hand and chasing the bad dreams away" (7.72). At least until the day she learned Sef was adopted. Having lost the only family she ever had, May deals with her loneliness by pretending that Sef is still with her in her old age.
At the end of the book, though, we get a big surprise when we learn that Mrs. Nightingale is actually Sef herself. Yay. And here's the thing: Until she reconnects with May, Sef in the present day isn't doing so great. She feels like baggage to her son and his wife and spends most of her time getting "immersed" (2.22) in card games and science fiction novels. Obviously, she's pretty lonely herself.
Nonetheless, though, she still retains the youthful qualities of the childhood companion she once was. As Lily observes, she was "a tall old lady with white hair woven into a silky coronet of braids—a well dressed old lady" (40.27) with green eyes that "looked so young" (20.39). Like May, then, Sef retains a bit of childlike whimsy to her. No wonder these two get along so well.
We're kind of disappointed that we as readers don't get to see May and Sef's reunion. But what we do get to see is still pretty awesome. On the morning of the family's party, Lily looks outside and sees "two old ladies in dressing gowns and slippers dancing closely across the lawn […] Lily thought Nan and Serafina's dancing sort of fitted: fitted the garden and the perfect lovely day she knew was coming, after all" (42.59). In a story full reunions, May and Sef's is one of the most joyous of all.