Short answer -
There is no "best Linux". You've got lots and lots of excellent choices.
You didn't mention what kind of processor you were going to use, how much RAM your system had, or whether or not you were familiar with Linux itself. These are all important considerations.
1. Get a reasonably powered PC, and install a common desktop distro (SuSE Linux, Fedora, Ubuntu). This will be your "development platform".
2. Install the common development tools you'll need (Gcc, PHP, Perl; Apache, X Windows, MySQL, etc) on your development PC. Familiarize yourself with them. Mock up your actual web app on it.
3. Get a USB stick.
Try a "small linux" (DSL, Puppy Linux, XPUD, etc).
Get it to boot from the USB stick, and see how it works.
4. Based on what you've learned in steps 1) .. 3), you're going to be much likelier to make informed decisions about how to design your final system, and your project will be much more likely to succeed.
IMHO .. PSM
If everything's going on a truly embedded device, then you might want to reconsider using MySQL, and use an "embedded database" (like Berkeley DB) instead: