Good news!  It is indeed possible to run an Xserver on an IBM 1420.  Specifically, I am running redhat 6.1 and am using frame buffering for the Xserver.  Much thanks to my research colleague Jamie Frey who showed me how to make the final necessary changes to the /etc/X11/XF86Config file.

The best resource I found to help me set this up is this page about Linux on Dells (they have the same chipset).   If you want to really understand what you are doing and perhaps seek a more optimal solution to run Xserver on your 1420, you should probably go directly to that page.  However, if you want a quick and simple solution, you should be able to do so with the following instructions (if you do figure a more optimal solution, please email it to me):

0)  These instructions will probably only work for systems that have frame-buffering.  I don't really know much about this except to say that redhat 6.1 has it, so this works for me.  These instructions will probably only work for laptops systems with a max res of 800x600.  Specifically, this page is intended for IBM thinkpad 1420's only.  Additionally, I don't even know if it will work with Linux's other than redhat 6.1.  If it doesn't work for you, there are some good general linux on IBM's and linux on laptop links at the bottom of this page.  Remember that the only thing that is important to get X running is your video chipset.  Which is why I was able to use the info about Inspiron's to get my X going.  The only trick was that the inspiron's support a higher resolution than the 1420.

1)  Make sure that /dev/fb0 and /dev/fb0current exist.  If not, create them like so:
    mknod /dev/fb0 c 29 0
    ln -s /dev/fb0 /dev/fb0current

2) Add "vga=788" to your /etc/lilo.conf file.  Type 'lilo' and then reboot.  If you don't see a little penguin in the upper right hand corner as you reboot, then I don't know what to tell you because you need to see that penguin in order for this to work.

3)  Download this FBDev X server.  Untar it using 'tar xvfz,' and copy it to /usr/X11R6/bin.  Make it the default server by setting the symbolic link for /usr/X11R6/bin/X point to it.

4) Download this FBDev XF86Config file and copy it to /etc/X11/XF86Config.

5) Reboot and startx.  Work blissfully.

If this doesn't work, I'm sorry.  Maybe these below links will be of benefit.  Or here are a couple more suggestions sent in by other people.
If it does work, pretty cool, huh?