D.C. Mayor Marion Barry yesterday nominated former City Council chairman Sterling Tucker to become chairman of the problem-plagued D.C. Board of Elections and Ethics.

Barry also chose Valerie K. Burden, a consumer consultant, to fill a second seat on the three-member board. Both nominees must be confirmed by the City Council.

The nominations come as the council is exploring ways to cure the District's chronic problems with its elections system, including possible reregistration of all voters and the postponement of all elections this year to give the board time to reorganize its operations.

Tucker would succeed current Chairman Albert J. Beveridge III, whose term expired in December, and Burden would succeed Virginia Moye, whose term expired more than a year ago.

Beveridge and Moye, who have continued to serve on the board, will remain until the council acts on Barry's nominations.

Jeannine Clark is the third member of the board.

Tucker, whose nomination has been expected for several weeks, served as vice chairman of the appointed City Council before home rule came to Washington and in 1975 became the first elected council chairman.

Tucker lost a close race for mayor to Barry in 1978. He briefly considered entering last year's mayoral contest, and later finished third in a bid to regain his old job as council chairman.

Burden, the daughter of Virginia Knauer, President Reagan's consumer adviser, is a resident of Georgetown and has been active in local Democratic party issues. She heads the Knauer & Associates consulting firm.

Burden holds a political science degree and a master's degree in business administration from the Wharton School at the University of Pennsylvania.