What can I do about a saddle valve that won't turn on?
Saddle valves are terrible as actual valves. They are a cheap/easy way to tap an existing water line, but that's it. After you use them a few times, you just can't rely on the needle to cleanly seal against the punctured hole of the copper pipe, and to cleanly come back out again.



In your case it also looks like there may be some galvanic corrosion on the valve, which might also be corroding around the needle and blocking flow.



Replace with a better valve

There are many ways to do this of course, but it basically means:


removing the existing saddle valve
cutting out the small piece with the hole in it 
putting in a tee
putting in a valve 




If you really can't get any water to come out of the compression valve, then this would be the best way to fix it (and probably never have an issue again). 

Use a real downstream valve

You can get away with using a saddle valve.. if you never actually use it as a valve. 

Basically, use the saddle valve but put a better valve downstream. In your case, you can use a 1/4" or 3/8" (I can't tell from the photo) compression valve on the flexible copper tubing -- assuming you can get water to come out of the saddle valve.