Virginia Lt. Gov. L. Douglas Wilder's Underdog Fund, criticized two weeks ago when it awarded its first grant to an overwhelming favorite in a House of Delegates race, gave its second grant yesterday to Emilie Miller, a Democrat running for state Senate in Fairfax County.

Miller, a Democrat who is considered a genuine underdog in her race, received $3,000. The lobbyist and mental health specialist said she was thrilled by her selection because "to get $3,000 at one time, I thought I would have to turn Republican."

Miller's Republican opponent is state Sen. John W. Russell, who by coincidence (he was attending a separate function at the same hour) wandered by in the midst of the Underdog ceremony in front of the Fairfax County government headquarters in Fairfax City.

Russell, a former mayor of Fairfax City who is running for his second term in the legislature, narrowly beat Miller in the 1983 election in the 34th Senate District, a band running east and west across the central part of the county.

As Russell looked on, Miller rapped him for having "a lack of respect among his colleagues and the media" and "unpredictability."

Russell's only retort was to wonder aloud why Wilder was supporting Miller "when Emilie's supporting {Attorney General} Mary Sue Terry for governor." Wilder and Terry are considered likely rivals for the 1989 Democratic gubernatorial nomination.

Miller responded that "no one knows" who she supports because "I haven't made my choice yet."

Although Russell is considered the favorite in the race, if only by virtue of his incumbency, Miller has raised more money. In their July 15 campaign finance forms, he reported $9,653 in contributions; she reported having received $15,262. Russell reported $3,592 in cash on hand; Miller reported $11,602.

Wilder founded the Underdog Fund after his 1985 election. It is intended primarily to benefit female and minority candidates for the state legislature. He was not present at yesterday's brief award ceremony.

The first recipient, lawyer J. Jack Kennedy, is running for the House of Delegates against a Republican woman in a Southwest Virginia district that is believed never to have elected either a woman or a Republican.