The street was named after George Downing, a man who served as Oliver Cromwell's Chief of Intelligence during the English Civil War. When the monarchy reclaimed power at the end of the war, Downing used his powers of persuasion to avoid execution and pledge his services to King Charles II. The first residence built on the site of 10 Downing Street was the home of Sir Thomas Knyvet, which was given to him by Queen Elizabeth in 1581. Knyvet's greatest accomplishment was arresting Guy Fawkes and fouling the terrorist attack on Westminster Palace. I guess that's worth a house.
Now, the first Prime Minister to reside at 10 Downing Street was Sir Robert Walpole. King George II offered Walpole the house as a gift in 1732, but Walpole only accepted the offer on the condition that 10 Downing Street become the official residence of the Prime Minister.
Because nearly all Prime Ministers preferred to live in their own homes, 10 Downing Street fell into disrepair over time. By the time Benjamin Disraeli was elected Prime Minister in 1868, the home was, in his words, "dingy and decaying." Disraeli convinced Parliament to pay to renovate the residence, although Disraeli had to pay out of pocket to refurbish the private rooms.