I want to increase the visual thickness of a line in ACAD 2006. I tried Format-Lineweight but it didn't seeem to have an effect.

Changing the Format-Lineweight will not affect lines that have already been drawn. But if you draw new lines, they will be drawn with the new lineweight applied.
 
If you want to affect lines that have already been drawn, you need to change the lineweight setting in your Layer Properties Manager. This will apply the lineweight to all existing lines the were drawn on the layer you adjust.
 
Or, if you just want to affect one single line, select it and then change the lineweight in your Properties palette.