Friends of the Rev. David Hilliard Eaton, the new president of the D.C. school board, say he's not a person who insists that things be done right now, or that his way is the only way. Board members who selected him as their leader insist that the lanky, low-key minister can bring a spirit of cooperation to the board.

Soon Eaton, who likes to spend time by the fireside in such peaceful pursuits as reading poetry and books on Eastern religions, will have his patience amply tested by the pressing problems of the D.C. schools.

Eaton, 49, insists he is up to the job. His leadership style, he says, will be one that served him well as a civil rights organizer and university administrator, as well as in his delicate current position as senior minister at All Souls Unitarian Church, which has one of the most active and diverse congregations in the city.

"I see myself as primarily having a general knowlege of the total spectrum," Eaton said yesterday. "I would then place people more knowledgeable than myself in areas where they have a certain expertise. I fulfill my responsibilities, but I get people who know what they're doing to help me. I have no problem delegating authority."

One of Eaton's strongest endorsements comes from Mayor Marion Barry, a longtime friend who attends his church. Barry, who has been at frequent odds with the board, said the election of Eaton (At-large) and two other new members to the 11-member body--R. David Hall (Ward 2) and Wanda Washburn (Ward 3)--makes him think he will have a "more positive dialogue" with the board from now on.

The arrival of Eaton and the others "made it a lot easier for me," said Barry, to decide last week to earmark an additional $10 million for the 1983 school budget. He noted that now there appears to be a new working majority on the board--one that includes the three new members and Frank Smith Jr. (Ward 1), Nathaniel Bush (Ward 7) and Linda W. Cropp (Ward 4). Smith, Bush and Cropp are members whom Barry supported for election in 1979.

"I feel closer to the majority philosophy of the current board than when I was on the board" in the mid-'70s, said Barry, a former school board president.

Though Barry praised Eaton as a person who "knows how to disagree without being disagreeable," he said he is also aware that Eaton has been criticized for not always exerting himself to the extent that some others think he should.

Barry said that Eaton himself "often points out that any time someone else says Dave Eaton is going to do something, he doesn't do much of it. But if it's something Dave Eaton says he's going to do, he does it."

Eaton refutes any suggestion of personal nonchalance by reciting his long record of community involvement. He says he is already setting up meetings with outgoing board president Eugene Kinlow (At large), school superintendent Floretta D. McKenzie, newly elected board Vice President Nathaniel Bush, and the board's executive secretary.

Members of his church who have known Eaton for years say he does follow through on his plans, though not always so promptly. "He takes time to get things done. . . . To some people this might be taken as laziness," said Janet Keenan, the assistant treasurer at All Souls.

Eaton said his immediate priorities as president include battling for more funds for the 1983 school budget and briefing himself on the coming contract negotiations with the Washington Teachers Union.

He said he would like to see the board establish a "research and evaluation committee" to do long-range planning of financial resources and educational programs for the schools. The schools, he said, sorely need to establish programs that will train students in specific fields, such as accounting and computer science, where he said jobs are expected to be plentiful in the '80s.

Like Barry and Walter Fauntroy, the District's nonvoting delegate to Congress, Eaton came to prominence in the city as a civil rights activist in the '60s. In 1969, he became the first black pastor of All Souls at 16th and Harvard streets NW, with a largely white membership that includes both city and suburban residents from varying economic levels.

The fact that Eaton has lasted so long at All Souls when his political views sometimes clashed with those of some church members is a testimony to his skills at human relations, said Noel Kane, who attends Eaton's church with his wife, City Council member Betty Ann Kane.

"I have never met anybody in my entire life who has the sensitivity toward other people that David has," Kane said of Eaton.

Eaton--who says his philosophy of education embraces the pragmatism of philosopher and educator John Dewey, the high aspirations for blacks of W.E.B. DuBois and the concentration on technical education of Booker T. Washington -- graduated from Mott Elementary, Shaw Junior High and Dunbar Senior High schools in the District. He studied philosophy and psychology at Howard University, and theology at Oxford University in England.

He lives in the elegant Northumberland cooperative apartments at New Hampshire Avenue and 16th Street NW with his wife Dolores, a teacher at Raymond Elementary School, and his daughter Claudia, a Howard University student.