How can I make the draggable window border thicker, without changing the appearance?

Change the following parameters in the appropriate Metacity XML file.

Ambiance's theme is located in /usr/share/themes/Ambiance/metacity-1/metacity-theme-1.xml

The folowing parameters change the width of the borders and consequently the width of the re-size area:

"<distance name="left_width" value="1"/>"
"<distance name="right_width" value="1"/>"
"<distance name="bottom_height" value="1"/>"

By default they are set to "1", but I found that changing them to "4" doesn't change the appearance very drastically, but makes it very much easier to re-size using the mouse.

You can't.

This is a three year old bug with no solution.

You either have to use a different theme or do as I have done and give up trying and use keys to resize windows.

Mini Howto:

Alt + F8 to start resizing. Use arrow keys to resize.

Pressing Return will save the resize, escape will revert you back.

The bug report is here and has hundreds of comments on it and thirteen duplicate bug reports on launchpad alone.