I can mix the ginger juice into the water and sugar mixture, but the juice will settle after a day. Is there no way to emulsify or stabilize the soda?
When I extract the ginger juice, I pass it through a fine-mesh strainer and discard the solids. I then let it sit for about an hour to allow the starch (white part) to settle. Finally, I pour that into another jar, leaving all the starch behind.
So, you let the juice sit to settle, then pour to another jar. If you let this new jar sit for a day or two in the fridge, will more particulates settle out? It could be that you haven't gotten the smallest ones that take the longest to settle, and then they are finally settling out once you mix up your drink.
Have you tried something like passing it through a paper coffee filter?
If you get rid of the particulate beforehand, you won't have to try make a colloid out if it, and you'll only be keeping the ginger extracts that are actually miscible in water.
If you really want to get the particulates homogenized permanently, you may have to research how to make food safe colloids. Emulsification is for liquids, and it sounds like what is settling out is not a liquid, so the emulsifier isn't working.
(I don't know how to do this with food; if these were bath bombs I'd tell you to try 1-2% polysorbate-80 by weight
)
If you are convinced that gum arabic should work, try increasing percentages of it in your ginger juice first. When you get a % that works for that you can move forward with further diluting with water and sugar.
Edit: I did some quick interweb searches. If I was set on doing this I'd try Xanthan gum, too.