How can I make this scanned signature into a solid lines to create an SVG?

This can be used to create a vector duplicate using the Trace-bitmap dialog.

Due to the coarse nature of the pen strokes and paper, it may need a bit of fiddly work to make it look smooth and natural. 

It's a difficult job to make it look good.

Using gimp, you can isolate the black and white signature. 

Using Inkscape you can then use Object>trace bitmap and experiment with various options.

But to get a really good effect, you would have to trace manually, using a calligraphic brush

Thank you both! I was able to trace with a calligraphic brush. But, it saves the image with a large invisible box around it. How can I just get the image only, saved as an svg?

Is the box a separate object which you can delete ? or are you talking about the bounding boxes - computers really only work with square framed images. You can have transparency in bitmaps such as PNG or paths etc in SVG, but the browser or art program you use tend to treat the outer limits of the shapes as an invisible frame.