No, they haven't changed the network's name to News By Clintons.

Chelsea Clinton earned an annual salary of $600,000 at NBC News before switching to a month-to-month contract earlier this year, sources with knowledge of the agreement told POLITICO. 

Clinton, who joined NBC News as a special correspondent in November 2011, was up for renewal or nonrenewal this year. Instead, the sources said, the network decided to keep her on the payroll on a month-to-month basis so the two parties could sever ties if Clinton’s mother, Hillary Clinton, runs for president.