Arlington County Board member Dorothy T. Grotos will become chairman of the five-member board Jan. 1 when Republican-backed independents take control of the board.

A coalition of Democrats, endorsed by the nonpartisan Arlingtonians for a Better County, has controlled the board since 1970.

The surprise selection of Grotos, 47, by the board's new majority was made at a recent luncheons meeting attended by Grotos, newly elected board member Stephen H. Detwiler and board member Walter L. Frankland J., who many had expected would become chairman.

"Frankly, titles don't mean that much," said Frankland yesterday. An outspoken critic of the Metro transit system, Franklin said he preferred being Arlington's delegate to the regional Metro board. "I was prepared to be board chairman and the Metro delegate but I wasn't overly anxious for both assigments," he said.

Frankland, who opposes completion of the full 100-mile subway system as too costly, will become vice chairman of the County Board.

"I thought seniority in the (Republican) Party should be a factor," said Grotos, who reportedly reminded Frankland during their discussions that she has received considerably more votes than he when both were elected in 1975.

"I've been vice chairman of the Republican Party and Walter just came in when he ran for the board," Grotos said. "One person can't do everything so it's a fair decision. Besides I haven't tended to come down hard on one or two issues like Metro."