I am running a full install of Slackware 12.2 quite well on an old IBM PC300 (Pentium 1 300 MHz Desktop--not laptop). Other than being a little slow, the only other problem I've noticed it that the Pentium I chip can't handle video. It does audio just fine.
Actually, it wasn't slow compared to my P3 XP box. It was just slow compared to other Linux boxes.
Right now, I'm using the box as a file server, but I used to use it as a webserver and ran a Wordpress blog off it for several years, until the database just got too big for the box to handle.
I suspect the laptop will run fine for internet access, except possibly for videos.
You might consider trying out some Live CDs first (remembering that any OS running off a CD drive is going to be slower than one running off the HDD just because of the seek time).