This thread has (unsurprisingly) been done a few times, such as here:
http://www.neogaf.com/forum/showthread.php?t=1314900
...But it's still fun to think about. The thing with putting a collection together for a mainstream audience is that you have to strike a balance between genres and core appeal, while also still being a little risky and trying to expose people to games they might have missed. Here's my list:
Crash Bandicoot (1 is iconic but I think 2 or 3 would be better to include)
Crash Team Racing
Metal Gear Solid
Final Fantasy VII (you kind of have to include it)
Final Fantasy IX (give people another opportunity to fall in love with it)
Castlevania: Symphony of the Night (again, you kind of have to)
Gran Turismo 2
Jet Moto 2 (synonymous with the PSX in my mind)
PaRappa the Rapper
Resident Evil (again 1 is pretty iconic but 2 might be better to include)
Tekken (3 is the safest bet I think)
Tobal 2 (be bold and put an import in there, especially when it's so good)
Tony Hawk's Pro Skater (I prefer 1 but recognize 2 is more widely loved)
Twisted Metal 2
Wipeout (tossup between XL and 3)
Einhander (shine on, gem)
Tomba (I played 1 at release and wasn't impressed but people love 2 a lot so I'm rolling the dice???)
Omega Boost (would be a good showcase game; has its fans)
R4 (would also be a good showcase game but is good on its own)
Driver (as close as it gets to a modern GTA on the PSX)
Hot Shots Golf (never played these on PSX but came to love them later in life and would love to go back; also good to diversify the lineup)
NFL Blitz (best home version of the game, right? Makes sense for it to be here)
Syphon Filter (not my jam but I know it has its fans)
Spider-Man (never played it but it's well-regarded and would have wide appeal; also good luck with licensing!)
Tenchu: Stealth Assassins (is this considered a hidden gem? I remember it being very difficult, hard to tell if it would hold up)
Bushido Blade (get that intense 2-player action)
Rival Schools: United by Fate (great fighting game for a PSX showcase)
Silent Hill (not my jam but I respect it)