My opinion is without a doubt biased, as I've only owned one kind of DS, but regardless, my vote goes to the DS Lite. I've played an original DS, but I've never owned one.
Here's what I like about the Lite:
-The screen brightness. The Lite's lowest brightness setting is brighter than my somewhat-dim GBA SP's screen. And the highest one makes it possible to play this thing in broad daylight.
-The smaller and lighter size of the system. It makes it very portable. I often take it to work to play on my breaks, and it fits inside my purse quite well, case and all.
-The larger stylus. The original DS's stylus is quite small - I'd run through those things like I run through guitar picks. I haven't lost one DS Lite stylus yet, though.
-The positioning of the stylus. It's very easy to get at, and stays in place when not in use.
Here's what I like about the original DS:
-It looks a lot more durable and hardy. I agree with the "DS Lite looks more breakable" point that some people mentioned. Whenever I imagine the Lite dropping and hitting the floor, I imagine its top half breaking clean off.
-Its more rounded form makes it easier to hold for long periods of time. Though it isn't as small and light as the Lite, its edges aren't as "sharp" as the Lite's.
-GBA games don't stick out the bottom of the system like they do in the Lite. It looks less clunky (though I find that the GBA cartridge sticking out the bottom of the Lite doesn't really affect gameplay).
I'm not sure if you can buy original DS systems new anymore... the only ones I've seen on sale are the Lites. You might be able to find an original DS at a used game store...