Standard rock band. Which member is coolest and why?
Tough question, but your use of the word “cool” (with its connotations of mellowness) rather than, say, “awesome” or “hip” puts the bassist over the top.
The only one we eliminate right away is bassist. The greatest bassist in rock history (IMHO, at least, Paul McCartney himself, will tell you that NONE of the Beatles wanted to play bass, which is why the inept, incompetent Stu Sutcliffe was kept around so long.
Despite all the drummer jokes, it’s occasionally possible for the drummer to be pretty cool.
A highly charismatic singer trumps the guitarist, but MOST of the time, the lead guitarist is the coolest guy in the room.
It’s always the bass player. He’s not flashy, he may not be sexy, but he’s cool.
To expand on that a little:
Drummer: insane
Guitarist: sexy
Singer: ego the size of a planet
Bassist: cool
For certain bands, you can switch the singer and guitarist.
Lead guitarist. He’s the real core. Would you rather be Pete Townshend or Roger Daltrey? Eddie Van Halen or David Lee Roth?
My answer is the product of two factors: first, I’m a musician myself and second, I came of age in the 1970s.
If you know anything about music in the '70s (other than disco, which is an abomination on this earth and shall not be spoken of), the guitarist was king. The pretty girl always got a crush on the guitarist. My daughter-in-law tells me that nowadays girls get crushes on djs. How sad.
So there’s that.
While an argument can be made for the difficulty of all the usual band instruments, and the difficulties of developing a really good rock voice, and also regarding which instrument is the canvas upon which the music is painted, I still have to come down on the side of the guitarist. As lead guitar, you can’t hide poor musicianship. You are right out in the front of everything and you’re judged by the quality of your inevitable solos.
Before the rotten tomatoes start flying, I fully agree that each instrument is powerful in its own way. I play bass and 'boards and sing, and I like to think I make a solid contribution. But popular music is almost synonymous with guitar.
Where’s the ‘keytar player option’?
I honestly can’t decide on either of those. I’ll tell you what though, I’d take Bonham over Page, or Peart over Lifeson, no question.
The thing is though that people identify with the vocalist. Studio musicians, or talented players looking for a gig, can always fill in for the instrumentalists. The band won’t sound the same without the vocalist though. That’s my choice for coolness.
Bass player*.
Maybe you don’t get the image, or the girls, or the magazine-covers, but you know (if you’re a good bass player) you bring da groove. Of course, root-only or root-fifth-only, or quarter note + two eights bass players are lame.
*The fact that I’m a bass player played no part in my choice. 
Hmmm, Lemmy. John Entwhistle. Phil Lynott. Peter Hook…I see a theme emerging.
Sid Vicious? ah, OK never mind. Carry on.
Bootsy Collins plays bass. I rest my case.
When you’re the bassist you get to play grooves like this one.
If you’re asking me who I like and think is “cool,” I’m a rhythm section guy, so drummers and bassists for me. I voted drummer today, although I may have voted bass another day. Now, if you asked me who popularly is perceived as “cool,” that would be lead singer or lead guitarist, of course.
ETA: Actually, reading over the comments, I think I should have voted bassist. But I’m glad to put an extra vote for the drummer, since they seem so far behind at this point.
Please. It’s the guitarist.
Actually, truth be told, it’s the songwriter(s).
Kim Deal and Kim Gordon. Bass players in cool bands. Each the coolest person in the cool bands.
I voted for drummer because of Charlie Watts - easily the coolest Stone!
True in those examples. But how about Jim Morrison or Robbie Krieger? Freddie Mercury or Brian May? David Bowie or Mick Ronson?
Sometimes the core of the band is the lead guitarist and sometimes it’s the vocalist.