I just made some ginger ale syrup for making home-made gingerale. It’s pretty easy! Just peel some ginger (use a spoon to remove the skin so you don’t waste the tasty flesh) and chop it up into thin-ish chunks, and put it into a saucepan with some water, some sugar, and other flavorings as you see fit (I used some pomegranate molasses, some coriander seed, and the peel and juice of a lemon). Then heat the water up and let the ginger steep for a while, then slowly bring it to a low boil.
Occasionally test the flavor, both to adjust the sugar level and to know when it’s gingery enough (don’t worry about extracting EVERY LAST BIT of ginger flavor, as that’s impossible and the point is to make something that tastes good, right?), and when it has a good flavor, use a wire strainer or the like to fish out as much solid stuff as you can. Then raise it to a moderate boil, and let it boil down until it’s 225-230F (don’t let it go any higher than that though, since at that point it starts to turn into candy).
Let it cool, and while it’s still warm and runny, strain it through a mesh strainer into a storage bottle of some sort.
To make the ginger ale, just mix some of the syrup with some soda water.
For bonus points, the chunks of ginger can be rolled in sugar and then put into an oven at 250F or so to dry them out a bit. And this is how you make candied ginger.