Analogue synth. 
As others have said, guitar is versatile and fairly easy to learn. People enjoy listening to guitar in all sorts of situations. I tried to learn it myself, but failed utterly. I am uniquely non-talented at music.
I have several friends who play the Irish tin whistle. They are tops in portability, cheap, and a lot of fun. I’ve been trying to learn the recorder. I’d love to have something very portable that I could play a few tunes on. (I’ve also failed at the harmonica.) Recorders are among the easiest instruments to learn.
And I’d love to play washboard, but I have no rhythm at all.
By the way, when you say you know “nothing about music,” what do you mean? I would say the same thing about myself, but I do know how to read simple sheet music, etc. Do you have any background at all?
Good bassoon players are worth their weight in gold, Blalron.
Go for the guitar, though. Bassoon players never get laid.
Gotta say harmonica…for many reasons. I was at a party awhile back and there were a couple of guitars and a drummer just jammin. (I play guitar)
BUT
Can’t tell ya how cool it was to whip out the ol’ harp (I keep one in the glovebox o’ my car) and start blowing into the mike they had setup. Everyone was like … “man that’s f***ing awesome”
It’s a cheap instrument, not that hard to learn and easy to carry.
Plus it is COOOL 
For making the neighborhood cats yeowl, go for the piccolo. VERY portable. Sweet sounding. There’s some music specifically written for 'em also.
Flute is good. I’m now into my 26th year as a flutist (or flautist if you want to sound pretentious) and while there is a fair amount of us, most seem to quit after high school/college. I rarely want for jobs.
Because LilMiss has taken up the violin, I’ve decided to learn it also. It’s pretty neat for both of us to learn at the same time (actually, she’s teaching me the fingering and I’m teaching her how to read music). She has banned me from bowing though. I’m not allowed to pick up the bow until I learn the fingering. Brat.
But for pure sexiness, go bass guitar. mmmmm… Most of the musicians I know are bass players and every single one of them… drool drool…
Not true. Don’t make me produce evidence.
(Percussionists seem to get laid the most, but in my experience, trombones and baritones are the best kissers).
Blalron - many good suggestions here already. Allow me to make a few additional observations:
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You don’t define “cool” - that somewhat limits posters’ ability to pinpoint an answer…
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Demonstrating technical competence, a broad musical catalogue (e.g., you know a bunch of songs) and the ability to swing and/or convey emotion is cool with any musical instrument. A mean set of spoons beats a bad violin every day.
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Guitar and piano are super common for a reason - versatile, used in most popular music forms, can be played while singing, comparatively easy to pick up the basics and play something that makes you feel like you’re making progress, but a lifetime’s potential for growth. There is a reason that most classically trained musicians, in addition to learning their primary instrument, must also know the basics of playing a piano. Same thing for popular music and the guitar - regardless of what you play, you need to know your guitar basics, just to speak the language of rock and blues…
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As a guitarist, I have stumbled into more situations where being able to pick up the guitar and get a group singing Beatles’ songs or whatever has been really cool.
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If being cool means being in demand, then solid bassists and drummers are always good to have around - having said that, there may be millions of guitarists and piano players, but relatively few are very good - i.e., ones that bassists and drummers really want to play with…
Hope this helps.
A girl goes on a few dates with some guys from a symphony. Late one evening she is talking about it with her room mate.
“The flute player couldn’t kiss at all, his mouth was puckered too tight all of the time. The trumpet player was a much better kisser but he slobbered all over me. Now, the French horn player wasn’t that good of a kisser, but I loved the way he held me.”
(Percussionists seem to get laid the most, but in my experience, trombones and baritones are the best kissers).
This is true. Looking back I did know a baritone player at one point who was a good kisser.
Damn near best kisser I’ve ever kissed played the banjo, tho.
The coolest of all instrument? One made of ice.
Lots of good advice in this thread, both serious and humorous.
I think the most important thing for you to decide is what do you think you’d like to play. Then give it a try.
My own personal experience is that I discovered I needed an instrument that I could enjoy playing solo, yet could produce a full sound. That means choosing an instrument that can produce chords, like a guitar or piano. (An accordian would work too, but I become homicidal when I hear an accordian.)
From third grade until I graduated from college I played the trumpet. I love the trumpet. But it is best as an ensemble instrument. I found that it wasn’t particularly rewarding to sit and play alone. The neighbors in the apartment next to me didn’t appreciate it much either.
After college, I picked up classical guitar. Took lessons for many years, until my daughter expressed an interest in playing piano. So I said, cool. I’ll do that too. So I shadowed her lessons for years and also learned how to play piano.
Now, if truth be told, I’m not very good at either the guitar or piano. But I love them both. And you don’t have to choose a typical classical or steel stringed guitar. You could go with a banjo or mandolin or ukelele if you want. And since portability is important maybe you could substitute an electronic keyboard.
So here I am, 25 years out of college and what am I playing? The trumpet again. (Well, actually a bugle… I’m an old fart marching in a drum and bugle corps putting on a full field show).
Life is a circle.
Choosing an instrument that stands out in your favourite type of music is a good way to go.
For me it’s bass guitar.
4 string bass…grr
5 string bass…ggrrr
5 string slap bass…Gggrrrr
Female 5 string slap bass player…hold me back!
Bass is fairly easy to learn, you get a feel for holding a rythym and counting to four (that part took me ages). There are not many bassists around so it’s easy to get a gig. You learn to play guitar at the same time without knowing it (bass guitarists are good at everything…and it really peeves some guitarists when the bass player can rip out a rocking solo on their guitar but they can’t play happy birthday on your bass).
You actually learn alot about music from any instument, which is the valuable bit, being able to transpose it to another comes with time. Pretty soon you’ll have all kinds of instruments hiding in corners. Practise and patience pay off as well.
Octraventral heebiephone, without a doubt.
Just make sure you have eight mouths.
Boldface Type wins. Definitely the coolest instruments.
Speaking (writing?) as One Who Has Been There, please, whatever you do, don’t try the violin. Hellishly long learning curve, even with dedicated practice; no social value; zero cool points; only really good for classsical music, some folk music and the sort of jazz not many people like.
I play guitar. It’s the only portable polyphonic instrument; great social value for singalongs and instant karaoke; can be a lead instrument or a rhythm instrument or a bit of both; has a role in classical, rock, jazz, folk and world music; once you’ve learned on an acoustic, you can play an electric, which opens up a million more possibilities in terms of sound, tone, effects and guitar synths etc.; can be very cool indeed; much easier learning gradient; very enjoyable; cheap to start (you can pickup a 2nd hand guitar for next to nothing; cheap to maintain (new strings every 6 months and that’s it).
I know you said you didn’t want to choose guitar because it’s too common. But once you’ve mastered the rudiments, you can find so many niche areas to specialise in, or choose a less-common type of guitar or playing style just for the added cool factor: 12 string, steel string, pedal, acoustic bass, double-neck, slide etc.
Finally, if you’re talking cool, I have no wish nor need to know anything about your sex life, but a man who has mastered the left hand guitar techniques of sustain, vibrato and tremolo is a man with a left hand that ladies can fall in love with. And that’s all I’m saying about that.
Another vote for this route. As you can tell by this link it is my instrument of choice. 
I have to argue the “always in tune” comment though. I have two Moogs that do not stay in tune until they’re warm.
The nice thing about the synth (or sampler) is,. if you start writing you can play almost any sound. My last track has a 70’s style chuckawacka guitar line. Everyone who has heard it so far THINKS its a real guitar, but it’s a sample.
[hijack]Continents drift, why not oscillators?[/hijack]
[further hijack]What? No Oberheim? No Chroma? You slacker! :)[/hijack]
Yes! I’ve been playing the trombone for 20 years now. Ok ladies make a line no need for pushing. Oh wait where’s everyone at?
The only thing I’ll caution about synthisizers is that you need to keep in mind that you won’t be able to easily go to a piano. Even though it’s still a “keyboard” , your hands won’t get the training needed to hit the keys on a piano with the same power and intensity that you’d need to produce the sounds correctly. While you can go from a piano to a keyboard, I don’t think you could do the opposite very easily.
Good luck, and let us know what you choose.