Mayor Marion Barry yesterday promised to nominate a new chairman of the D.C. Board of Elections and Ethics this week after current Chairman Albert J. Beveridge III threatened to quit in frustration over the delay in naming his replacement.

Barry, already embarrassed by his months of difficulty in recruiting a new board chairman acceptable to the City Council, delayed a high-level staff meeting for half an hour to meet with Beveridge and persuade him to stay on a little longer.

Beveridge said later that he had been ready to quit because he was under the impression that "nothing was being done" by Barry to find a new chairman.

"I was concerned the process was taking longer than what was best for the board," he said. "I was not fully informed. Now I am fully informed . . . and I think things are back on track."

Annette Samuels, the mayor's press secretary, said Barry told Beveridge that he was "close to finding someone" for the job and asked Beveridge "if he could be patient just a little while longer."

The mayor's long-frustrated effort to find replacements for Beveridge, whose three-year term expired last December, and for Virginia Moye, another member whose term expired more than a year ago, has kept the board from dealing with a wide range of problems.

High on the list is finding a permanent replacement for Teddy Filosofos, who resigned last October after a brief stint as the board's executive director.

The three-member board also is in the midst of trying to overhaul its trouble-plagued voter registration system and prepare for the fall elections--assuming the City Council doesn't postpone the elections, as some have suggested, to give the board more time to deal with its problems.

Barry nominated former City Council Chairman Sterling Tucker Feb. 23 as board chairman, but the council's Government Operations Committee rejected the nomination March 10. The committee also killed the nomination of Valerie Burden for Moye's seat.

Since then, the mayor has looked at scores of possible candidates for the chairman's post, which pays up to $26,500 a year. In many cases, the mayor wasn't satisfied with the candidates' qualifications, Samuels said.