Final answer:
Ishikawa Goemon is the correct answer; he is a legendary thief from the Sengoku period whose family was subjected to a brutal execution. His tale resembles a Japanese version of Robin Hood.
Step-by-step explanation:
The name of the gentleman thief whose family was boiled alive during the Sengoku period is A) Ishikawa Goemon. Goemon is a semi-legendary character in Japanese folklore, celebrated as a heroic bandit who stole from the rich to give to the poor. His actions have often been compared to those of Robin Hood. Goemon's end is a tragic tale of martyrdom where, according to folklore, he tried to assassinate the warlord Toyotomi Hideyoshi to avenge the death of his wife and was subsequently boiled alive alongside his young son.
Accounts of Goemon's life are a mixture of fact and fiction, and while some details of his story may be embellished or mythologized, he remains a popular figure in kabuki theatre, jidaigeki (period dramas), and various other cultural depictions. His legendary status was solidified during the Edo period, after the Sengoku period had ended.