It seems that Becket was appointed Keeper of the Works for the Tower of London (custos operum turris).
Richard de Luci (Chief Justiciar) was Constable of the Tower at the time
1155 Thomas Becket, keeper of the works at the Tower, carries out repairs.
Douglas Knoop; Gwilym Peredur Jones (1933). The Mediaeval Mason: An Economic History of English Stone Building in the Later Middle Ages and Early Modern Times. Manchester University Press. p. 4
already, in the twelfth century, there is an anticipation of the nineteenth, in William Fitzstephen's description of Thomas A Becket's repairing of the Tower of London :- With wonderful speed accomplishing so great a work between . . . Easter and Whitsuntide, with so many smiths, carpenters and other workmen, working so vehemently with bustle and noise that a man could hardly hear the one next to him speak.'
Materials for the History of Thomas Becket (Rolls Series), III., 19-20.
James Craigie Robertson. Materials for the History of Thomas Becket, Archbishop of Canterbury (Canonized by Pope Alexander III, AD 1173). Volume 3. Rolls Series Cambridge University Press. pp.19-20 . ISBN 978-1-108-04927-6.
Translation, extract from
Gourde, Leo T. (1943), "An Annotated Translation of the Life of St. Thomas Becket by William Fitzstephen". p 27
Thomas repaired the Tower of London, the seat of the
Kingdom, which had become dilapidated. This big task proceeded
with such speed, that the work was completed between
Easter and Pentecost. There were so many carpenters and
other artisans employed that they had the hardest time to
speak to one another and understand one another amidst the
din and noise.
John Whitcomb Bayley (1830). "The Office of Constable". The History and Antiquities of the Tower of London... Jennings & Chaplin. pp. 614–. Retrieved 17 September 2013.
John Whitcomb Bayley (1830). The History and Antiquities of the Tower of London. Jennings & Chaplin. pp. 8–.
Andrew Lythall (October 2009). How Did the Murder of St. Thomas Becket Affect the Relationship Between Church and State in England 1170-1215?. GRIN Verlag. pp. 5–. ISBN 978-3-640-45817-2.
Samuel Astley Dunham (1834). A History of Europe During the Middle Ages. Longman, Rees, Orme, Brown, Green & Longman, Paternoster Row, and John Taylor, Upper Gower Street. pp. 172–.
C. St. George (1830). A civil and ecclesiastical history of England, to 1829. pp. 100–.
Joseph Wheeler (1842). A Short history of the Tower of London; with a list of the interesting curiosities contained in the Armories and Regalia. T. Hodgson. pp. 7–.
Wikipedia Tower of London
Wikipedia Constable of the Tower
People of the Tower.