MIDLAND, Tex. — President Trump does not know Mike Conaway.

A Republican congressman from a long brush stroke of West Texas, Mr. Conaway recalled meeting with him at the White House with other House Republicans. And he has shaken hands with Mr. Trump, a “standard, 500-people-on-a-rope-line, shaken-hand kind of thing.”

“He wouldn’t know me from third base,” Mr. Conaway said.

Whether he has exchanged pleasantries with the president may not have mattered before, but it does now. Mr. Conaway is taking over the House Intelligence Committee’s investigation into Russian meddling in the presidential election. He is replacing Representative Devin Nunes, the California Republican whose suspiciously cozy relationship with Mr. Trump derailed the inquiry before he was ultimately forced to step aside.

Mr. Nunes’s missteps have thrust Mr. Conaway, a low-profile, old-guard Republican, into the spotlight as the accidental heir to a potentially explosive investigation swirling around a president from his own party. He said he would dutifully pursue an extraordinary allegation that he appears to find dubious: that Mr. Trump’s associates may have worked with Russian officials to disrupt the election.

The vow of a vigorous investigation may be reassuring to some, but in Mr. Conaway’s district, many dismiss it as an effort to disparage Mr. Trump. On top of that, Mr. Conaway has never been under the glare of national scrutiny.