They didn't change their name--it was changed for them when
their patron changed or when his job changed. In fact their name
changed several times: From the Lord Chamberlain's Men to Lord
Hunsdon's Men when the Lord Chamberlain died and his son Lord
Hunsdon took over patronage of the company, from Lord Hunsdon's Men
to the Lord Chamberlain's Men when Hunsdon became Lord Chamberlain
like his father, and from the Lord Chamberlain's Men to The King's
Men when King James I took over patronage of the company. The
partners in the company had nothing to say about it.