I don’t think it’s biologically possible for Elizabeth Cocke to be his mother. She married three times: (1) Robert West, (2) Sir Robert Oxenbridge, and (3) Sir Richard Lucy.
She married Sir Robert Oxenbridge before 1603. Her father’s will, dated 22 Sep 1603, states:
Item whereas ... There maye perhapps arise some doubte or question upon the assurannce and conveyannce heretofore made by me to my daughter Oxenbridge uppon her marriage with Mr Robert Esquire [sic] deceased … do hereby further … bequeathe the said Mannor of Broxborne in the said Countie of Hertford … after the Decease of Dame Ursula my wife unto Sir Robert Oxenbridge my sonne in Lawe and to Dame Elizabeth his wife my daughter … Then to Harry Lucye sonne of my daughter the Ladie Lucie …
(His daughter Lady Lucy, was Frances, the wife of Sir Edmund Lucy of Kingswood, Hertfordshire.)
Sir Robert Oxenbridge died and Elizabeth married Sir Richard Lucy before the Visitation of Warwickshire in 1619, which records in Latin (at page 261):
Elizabeth, daughter of coheir [of Sir Henry Cocke]
wife of Robert West son of Lord Delaware
Robert Oxenbridge of Husbandes in County South’ton knight 2nd husband
Richard Lucy knight and banneret 3rd husband
I think we can confidently say she wasn't the mother of Sir Kingsmill Lucy, who was born in about 1649-50. His mother was presumably Sir Richard Lucy's second wife, Jane Chapman.
Historical Gazetteer of London Before the Great Fire (1987) on British History Online says:
... when Thomas Chapman II died in 1626 his heirs were his 3 daughters Rhoda, Rebecca, and Jane. These daughters inherited equal shares in the property (22, 25-6), which was not physically partitioned between them until after the Great Fire. Differing accounts of their marriages and children are given in different secondary sources. The following explanation seems the most plausible.
Rhoda, the eldest, born before 1620, married firstly Thomas Hussey of Wormley, Herts., who died before 1645, and secondly, in 1646, Ferdinando, Lord Fairfax of Cameron, who died in 1648 ...
Rebecca, the second daughter, married firstly Thomas Playters (d. 1651), son of Sir William Playters, bt., by whom she had no children; secondly Richard Lucy, esquire (d. bef. 1669), son of Francis Lucy, M.P., by whom she was probably the mother of Platiers Lucy, who was under 21 in 1671; and thirdly, before 1671, Sir Rowland Lytton, bt., of Knebworth (d. 1674), by whom she had a daughter Rebecca.
Jane, the youngest daughter of Thomas Chapman II, married Sir Richard Lucy, kt., bt., of Broxbourne, Herts. (d. 1667), as his second wife, in or before 1645, and was probably the mother of his son (Sir) Kingsmill Lucy. Sir Richard Lucy was the uncle of Richard Lucy, esquire. Jane seems to have died before 1659.