I got curious about this myself, so I went to look in Stenson's bio of him. Much to my chagrin, he does not give a specific date. On page 315, he merely states that Lightoller retired "at the beginning of the 1920s".
Stenson does, however, leave some hints. He states on page 292 that at the time of Lightoller's retirement, his two eldest sons were in their teens, who were born about 1904 and 1907 (not sure of exact dates on these). So, 1920 would be about the earliest time for him to have retired, as the younger son would have been 13 then. On page 293, Stenson also mentions the jobs Lightoller had after his retirement and his trouble finding suitable employment because of his age, stating that he was "nearing fifty". Lightoller turned fifty in 1924, so I suppose his retirement had to have taken place between 1920 and 1924, probably closer to 1920 than to 1924.
It seems odd to me that this is not already generally known. Seems strange that Stenson did not pin this date down when he wrote the Lightoller bio.