I was born in Pity Me and still live quite nearby, despite my childhood in Harrogate. There were lots of suggestions as to the place name, some of which revolve around French monks at nearby Finchale (pronounced Finkle) calling it Petite Mer, or Petite Mere, referring to areas of boggy ground. The OED also feels it’s just possibly a reference to peaty ground where things were difficult to grow, or Pity is a reference to the local mining history as in ‘pit’. There’s another suggestion that the monks carrying Saint Cuthbert’s coffin dropped it at Pity Me, where the Saint begged them to continue to carry him onto Durham as a final resting place.
Pays your money, takes your choice - as the saying goes.