I have [url="http://imgh.us/Chess_diagram_bw.svg"]this image of a chessboard[/url]:



The dark squares are formed with lines passing diagonally across the board, hidden by the light squares. What I would like to do is make each "dark square" selectable as a square, as the light squares are. Of course I could create white square behind each dark square, but that would be tedious and the resulting squares would be difficult to select.

You cannot "convert" lines to squares, nor have square spaces between the white squares.
May I suggest you to use the tiled clones panel,
or the duplicate (Ctrl+D), group (Ctrl+G), and align and distribute panel (Ctrl+Shift+A) options,
to create a simple chessboard.
To add a pattern to the squares,
you can create a fill pattern with lines, or use some sort of masking/clipping.
May use cloning as well?
Also taking in consideration of the final display size is important, otherwise it can end up not screen-fitted and edges displayed badly.
By the way, how do you want the dark squares to be selectable?

How can I create a fill pattern with diagonal lines (as shown in the diagram) that will look exactly the same on every square?

The current pattern doesn't seems to repeat on each square, the top left lines seem thicker ones.
If you combine them together and add a fill to them after, you could add a look as now.
If you add a fill to one square only and duplicate/clone it, it would produce each square to have the exact same fill.
I wouldn't go with the pattern option though, you can have more control over the pattern manually in my opinion.
Will post some example how I would draw this soon.

Here is how I would draw a similar one to yours:

The squares are clones of a parent, so you can adjust the colours and such.

But, transparent fills for the dark squares may not be the best idea,
so here are some variants on a chessboard with an opaque fill:
Some variants on a chess board

That board is exactly what I want. Will you please give me permission to license it under CC-BY? And is there a simple way to make all the dark squares' fills white?

If you open it in inkscape you will see tiles on the right that are parents of the clones.
You can do whatever to them, the clones are following.

If you licence it under cc-by, you would claim it as your own work.
As I posted it here it's more of a public domain now, if you don't draw your own I guess technically it can't be cc-by.

I missed the line pattern ordering for the dark squares, guess it was just an example how it would look
if it was drawn with the mentioned tools.

Still it's beyond me how anyone would benefit of such a basic drawing.
If someone has difficulties with making such design,
chances are big there would be more problems, and a messed up file as a result.

Are you saying you have released that chessboard into the public domain? Do you renounce all copyright claims to it?

Well it wasn't uploaded to openclipart just my dropbox and made it public.
I do renounce any copyright for any material alike I post here.
Still after that, noone can have copyright claims on it as it's public domain.