In the program Teralogic Texture Maker (or TTM) you would
"load buffer" with the graphic
then "mix buffer" several times

Just hit the button that says "make texture seamless" to get a seamless tile.
By playing with the settings in the "finished texture" window you can get different results.

In the program 20/20 it is simply
image/tile effect

This produces a really nice seamless tile.

In the program Paint Shop Pro 5.0:
use the color picker to get a color from the original
make this the background color (hit the double headed arrow to put this color as the color swatch on the bottom)
Image/add borders
put a check in symmetic and type in 100
use the magic wand to select your border
selections/invert
selections/convert to seamless pattern

You needed to pick a color that was in the picture because it took some of the border you added and blurred it in with the edges to make a seamless tile that is not totally seamless.

Another method in PSP is to make your selection about a 100 px in from the edges using the rectangular section tool then go to
selections/convert to seamless pattern

This will give you a different result.

Combining the two ideas above gives this effect.

It is done by first making a copy of the image (edit/copy and edit/paste as new image)
On the original add borders like before and select the borders
Get the flood fill tool and set it to "pattern"
set the pattern to be the new image
flood fill the border with the image
selections/invert
selections/convert to seamless pattern.

Using Kai's Power Tools version 3 (version 5 will not work completely in PSP5) and the "seamless welder"
on the last tile above using the whole graphic and default settings

Or you can go back to the original and make a rectangular selection inside - pretty much the same as using PSP 5 itself, except that you can make your selection a bit closer to the edge.
this tile is what results from that kind of a selection and having the seamless welder set to "seamless" instead of "reflective".
Remember after you have applied the filter do not unselect the area, but go to edit/copy then edit/paste as new image. If you forget to do that, you will save the whole image, and not get a seamless tile.

Same methods, but set to "reflective" makes this tile.

Using the FM tile tools plugin "roll image" help files here
you can move the edge of the graphic into the center and use the clone, smudge, and push tools to get rid of the line. This method requires more time and skill, but allows you to make seamless in only one direction if you wish. (The DC Special plugins "edit do" and "edit undo" will also move the edges in, but moves both edges into the exact center. The FM tile tools give you more control. They also include a seamless tile maker.)

This is my preferred method of making seamless tiles as it preserves all of the original graphic and the original size. Here are the results