How can I build a custom distribution for running a simple web browser?

Many distributions have some facility for a minimal install; essentially where you manually select only those packages that you explicitly wish to install. Debian has this ability and would be a better choice, in your situation, than the other obvious minimal contender, Arch Linux.

Arch's rolling release status may provide a level of ongoing complexity that you wish to eschew. Debian would provide the simple, minimal base you are looking for plus offer stability. There is a blog post on using Debian as a kiosk that may offer some helpful tips.

For a browser, as beav_35 suggests, Uzbl is a good choice. My recommendation would be Vimprobable, a WebKit browser that is scriptable, keyboard driven and can be controlled effectively over SSH.

As a window manager, I would recommend dwm: at less than 2000 SLOC, it is extremely lightweight and can be easily configured for a kiosk-type setup.

First of all, you might not want to reinvent the wheel... There are several kiosk-focused distributions. One of those might save you a lot of work.

Second, if you are familiar with Kickstart on Fedora and RHEL(or CentOS or Scientific Linux), you can use the Fedora tools to make your own spin of any of those distributions. The documentation is a bit scattered, but start here.

Start with a minimal distro such as Arch Linux, then install the packages you need. For a simple web browser try Uzbl.