Lobus Caecilius was a man who lived in Pompeii before Vesuvius erupted on 24 August 79. Husband of Metella and father to Evelina and Quintus, Caecilius was a marble trader with political ambitions; it was to impress Lucius Petrus Dextrus, high priest of the Cult of Vulcan, that he bought the Tenth Doctor's TARDIS at the market, intending to display it as "modern art".
Along with his family, Caecilius was spared from the devastation of Pompeii by the timely intervention of the Doctor. By early 80, Caecilius and his family had resettled in Rome, and adopted the Tenth Doctor and Donna as their household gods. (TV: The Fires of Pompeii)
The Twelfth Doctor remembered Caecilius when seeing himself in a mirror for the first time, having seen his face somewhere before. He suggests, that he at least more than once had taken a new face from someone he had seen before: "I never know, where these faces come from." (TV: Deep Breath)
Behind the scenes
The name game
This character's surname was — by Russell T Davies' admission in The Italian Job — derived from books in the Cambridge Latin Course, elementary Latin books about real, historical people. For this reason, it is frequently asserted that this character is meant to be a "celebrity historical character" in the vein of Charles Dickens or Agatha Christie. Many people, including at least one official merchandise licensee, believe that this character is supposed to be the historical Lucius Caecilius Iucundus.
Though both plausible and beguiling, it's simply not the case. The script actually names him Lobus Caecilius upon his introduction to the Tenth Doctor; the words "Lucius" or "Iucundus" are never used in connection with Capaldi's character.
Finally, careful attention to The Italian Job reveals that the production team merely nicked the names from Cambridge Latin Course books. As Phil Collinson confirmed, "It's all in [the Cambridge Latin Course] — in terms of the names". But it was not the authorial intent that this character or his family be an attempt at recreation of historical figures. Instead, Davies characterised the use of the name "Caecilius" as a gag, supporting the many other broad Latin gags in the episode, similar to those found in his beloved Asterix comics.