Are you aware of any famous recent examples (possibly supported by an article) where the US President appointed a cabinet member because of his good relationship with the opposition?
President Obama nominated 3 Republicans to his Cabinet, but ultimately only two served. They are Robert Gates (Secretary of Defense, originally appointed by G.W. Bush and retained), Secretary of Transportation Ray LaHood (Representative from Illinois), and Judd Gregg (Senator from New Hampshire and former Governor). Gregg withdrew his name from consideration in February 2009. There is a press conference from January 2009 where he alludes to some of the motivation behind those choices being an attempt at reaching across the aisle:
You know, when I made a series of overtures to the Republicans — going over to meet with both Republican caucuses; you know, putting three Republicans in my Cabinet, something that is unprecedented; making sure that they were invited here to the White House to talk about the economic recovery plan — all those were not designed simply to get some short-term votes. They were designed to try to build up some trust over time. And I think that as I continue to make these overtures, over time hopefully that will be reciprocated.
Turns out, the PolitiFact fact check of Obama's claim put together a pretty good list for this question.
President George W. Bush appointed Norman Mineta as Secretary of Transportation. He was President Clinton's Secretary of Commerce for a period of about 6 months, but went on to be the longest-serving Transportation Secretary under President Bush. This was also an effort at bridge building, Mineta was an informal adviser to Al Gore who Bush just beat in an election that involved lawsuits decided by the Supreme Court. During the announcement, Bush introduced his nominees and then allowed each to make a short statement, and Secretary Mineta said this:
As you have said time and time again, Mr. President-elect, the challenge before all of us as Americans, regardless of party affiliation, is to find those areas where we can build bipartisan consensus on the policies and the programs that are needed to move this nation forward.
President Clinton appointed William Cohen (ex-Representative and Senator from Maine) as Secretary of Defense during his second term, replacing Secretary William Perry who announced his decision to step down shortly after Clinton's re-election. Clinton said the following when announcing Cohen to the cabinet position (from here, ~06:12 into the video):
Bill Cohen is the right man to build on [Perry's] achievements, to secure the bipartisan support America's armed forces must have and clearly deserve. [...] While serving the people of Maine he has also served every American through his determination to find common ground on difficult issues.
President Reagan appointed at-the-time Democrat William Bennett as Secretary of Education, who changed party affiliation the following year and has remained Republican since. I can't find any source about the announcement of Bennett's nomination from the administration as to the motives behind choosing a Democrat, but the Washington Post covered the defection.
President Carter appointed James Schlesinger (former Secretary of Defense under Nixon and Ford) as the first Secretary of Energy. You can see the announcement here, but it's not clear to me right now if it was an action directed at a wider population of Republicans, or if they just happened to agree on energy policy. From here:
James Schlesinger impressed candidate Jimmy Carter with his brains, his high-level experience (he'd run the CIA, the Atomic Energy Commission and the Defense Department), and with secrets regarding the defense spending vacillations of his old boss, Gerald Ford, just in time for the presidential debates. After the election, Carter hired Schlesinger as energy secretary.