The answer is William McKinley, who was president from 1897 until
September 1901 when he was assassinated shortly after the beginning of
his second term.
Here is an authoritative source for the information:
"But [what] McKinley noted was that the number of American newspapers
were tripling in circulation between roughly 1890 and 1910. What he
also noticed was that there were new associations of the press, such
as the Associated Press which begins at this time or the New York Sun
Syndicate which became a very powerful newspaper group. And McKinley
decides that these people are going to get news someplace, why not
from him? So he is the person who sets up the first press room in the
White House. He gives a place in the White House where correspondents,
who are now welcome, can do their work."
PBS: The American Experience: Interview Transcript: The First White
House Press Room
Here is a link to a brief biography of President McKinley:
White House: Presidents: McKinley
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"white house" "press room" first
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