In “To Kill a Mockingbird”, it’s claimed that many local families can trace themselves back to one side or the other at the battle of Hastings. Are there really any families from the Saxon side that maintained their prestige to modern times?

To my knowledge there are not and there seems to be no reason to believe there are. This is in part due to the very fragmented records that we have of even the Norman participants in the battle.

In short though, the claim in the book seems extremely dubious. England at the time had few prominent families. Three deserve special recognition. The King and his family, the Sons of Godwine, Earl of Wessex, Harold The other major family at the time were the grandsons of Leofric, Earl of Mercia, Edwin and Morcar.

The House of Wessex (the native ruling dynasty) were also still around, but they held little power compared to these two other families.

All of these families would be displaced by the Norman conquest.

King Harold and most of his siblings died in the Battle of Hastings (Wulfnoth survived as a prisoner of the Normans until his own death in the 1090's). Another of his siblings, Tostig, died in the Battle of Stamford Bridge. Harold left several sons, but they all disappear from history at various points.

Edwin and Morcar later rebelled against William the Conqueror. Edwin died fleeing in the aftermath of the rebellion's failure. As for Morcar it is unclear what happened to him. He was supposedly released on William's deathbed but then imprisoned by William II, he then disappears from records.

The House of Wessex managed to last a little longer but they did not participate in the Battle of Hastings. Eadgar the Ætheling was named king after Hastings but never crowned, and he did not dispute William's coronation. Eadgar got involved in a variety of disputes between the Normans and their neighbors in Scotland, Frisia, and the king of France. William of Malmesbury says that he eventually wound up in the Holy Land on pilgrimage after the 1st Cursade (some accounts indicate he participated in it, but this seems unlikely) and was offered places to stay in court in both the Byzantine Empire and the Holy Roman Empire.

His sister Margaret married the King of Scotland and their daughter married Henry I of England. She started numerous charitable works there, became canonized as a Saint after her death, and several of her children became kings of Scotland, and a daughter married Henry I.

Of the main participants at the Battle of Hastings on the English side little is known of their eventual descendants. Many perished in the battle or in the years afterwards, and their families vanished into obscurity.

Sources:

(Primary)

William of Malmesbury: Gesta regum Anglorum

William of Malmesbury: Historia Novella

(slight nod to Orderic Vitalis and his Historia Ecclesiastica who purported that Eadgar went on Crusade)

(Secondary)

Betty Hale. "History of Prince Edgar & his Claim to the English Throne"

Donald Henson, "The English Elite in 1066: Gone but not Forgotten"