According to her proof of age made in 1295, Lucy was born on Friday 24 March 1279 at Kilton, a manor and castle that belonged to her grandfather, Sir Marmaduke de Thweng, in the Cleveland district of North Yorkshire. She was baptised there the following Sunday at St Peter's Chapel, by her grandfather's chaplain Alan. Those later remembered to have been present were her grandmother Lucy, Marmaduke's wife, ‘Margery’ de Brus, who was probably her great-aunt Margaret, and Sir Richard de Thweng, probably her great-uncle and Marmaduke's brother. Also present were Peter Marshall and Richard ‘le Estivor’ or ‘Estywur’, who may have been members of the Thweng household. Lucy's father, apparently not present, was Sir Robert de Thweng, eldest son of Sir Marmaduke.
At the time of Lucy's birth, the Thweng family had risen, over the past century, to the peak of its prosperity, chiefly through marriages to heiresses. By c.1230, the family held the manors of Thwing and Lund in the East Riding, and Kilton, Kilton Thorpe and Kirkleatham in Cleveland. Lucy's great-grandfather, Robert III, acquired some notoriety in the 1230s for his resistance to papal provisions, but it was the marriage of her grandfather, Marmaduke, which was of particular significance for the family in general and for Lucy in particular. His wife was Lucy de Brus, third of five daughters of Peter de Brus (d.1240), a neighbour in Cleveland and head of what was probably the leading family in northern Yorkshire. Marmaduke and Lucy were married in 1242 when Robert III confirmed that he had granted Marmaduke the manors of Kilton and Kirkleatham to dower Lucy, and her maritagium may have been the manor of Little Moorsholm, near Kilton and Kilton Thorpe. Marmaduke had succeeded Robert by 1257 when he had an extensive grant of liberties from Henry III. By August 1260 he had joined the royal household as a banneret and in 1264 he was one of the royalists captured at the battle of Lewes, his ransom being 700 marks (£467). In 1272, his brother-in-law, Peter de Brus, died without issue, and the extensive Brus estate was divided between his four surviving sisters and their husbands. Agnes was married to Walter de Fauconberg, Margaret to Robert de Roos, and Laderina to John de Bellew.