Strength to Weight Ratio -- Units

N-m/kg. All of these derived units employing Newtons are clumsy, being based on a measurement of acceleration. But it's the short version.
I don't try to derive them backwards, to what they actually mean. You can do that once for your own understanding and then be done with it. The most common and useful way to use measurements of specific strength is in comparisons between materials, in which case, units don't matter.

I've seen this come up a few times in aircraft modeling fora, where they were quoted without dimensions.
Which always bugs me -- three WHAT?
Do you know of any prevalent systems in good old 'merican units, that you might see quoted for oak vs. fir in a lumber yard, or by a gray-haired engineer? Somehow I don't think that lbf-ft/slug would be quoted very often...

The true SI version uses Pascals. It's Pa/(kg/m^3). I don't remember seeing it used, but then, I don't read a lot of academic engineering white papers. The Newton, "commonly used metrics" version shows up quite a bit.
Do you really want to go there? The concept is simple: it's strength per unit area, divided by density. But the units can get strange.
Strength in pounds per *square* inch over density in pounds per *cubic* inch.
The simplification gives you very weird units. The reduced unit is inches.
As for wood, the _Wood Handbook: Wood as an Engineeering Material_ book from the Wood Products Laboratory only mentions it twice, and they don't discuss units. BTW, if you're interested in wood and you don't have that 509-page book you should download it. It's free. Then get R. Bruce Hoadley's _Understanding Wood_. It's not free but it's worth a bundle if you're interested in wood.
Once again, you may want to go through the exercise of deriving it *once*. Then you'll see why the units are weird. Then you can forget about the units and just use the numbers for comparisons.
After all, the units for Young's Modulus are very weird, too.