Over at Idolator, I got an answer to the question that was troubling me: Why do a cover of the Foo-Fighters’ “Best of You”?  The answer, with some help from commentator MosH8ed – because of a possible beef dating back to the Foo Fighters' cover of “Darling Nikki”.  A quick google-search gets us this:

But ask him if he's heard the Foo Fighters' version of "Darling Nikki" and Prince, who a minute earlier said he never listens to the radio ("When I want to hear new music, I go make some"), replies by describing a Hawaii DJ's response to the Foos' cover. The DJ wondered if Prince had heard it -- then said he couldn't care less if he had. "Just no respect," says Prince. "I wonder if that's the kind of thing the FCC would like to clean up, too."

So...does he like the cover? "No! I don't like anyone covering my work. Write your own tunes!" He says he got up in R&B singer Ginuwine's face for bungling the lyrics in a 1996 version of "When Doves Cry." "I was just busting on him to bust him, but I was a little serious: Have some respect, man. If anyone tried to cover 'Respect,' by Aretha*? I would shoot them myself!"
- Entertainment Weekly '04

Couldn’t find the actual Entertainment Weekly article, but its text is here.

Some more on that cover of Darling Nikki (this one has no source, just a comment on a Prince fan site):

Prince let Dave Grohl know of his displeasure of the cover of "Darling Nikki" and Dave had no idea. He apologized to Prince and said he was doing it honor of him. It was put out as a b-side to a single on import and only a first pressing exists as it was not recollected by the record label after Dave asked it to be pulled.

KROQ in L.A. got a hold of it and was playing it and it made it to the number one most requested but Dave called the station and let him know if they could stop playing it due to Prince's displeasure as to not upset him. They (KROQ) let it be known at the end of the week, due to Dave's request, it would stop being played.

Whether true or not, it certainly seems plausible that Prince decide to take advantage of the biggest stage in the world (the Super Bowl game, with its 140 million viewers) to play the hell out of a Foo Fighters song, much better than they could ever play it, and pay back the favor for attempting to cover “Darling Nikki”.