Well there's not much more to say than in general go with a double walled rim. Since that describes any rim outside the bottom of the barrel, that's not going to help you much. You might take some time to look around at online reviews to see what others like and actually spend the money for a new rim. Even if this is a rim issue (more on that later), why would you ditch one fouled up rim to get another used and abused rim? It doesn't make sense.
HOWEVER, I suspect that the real problem is not your rim at all but in the spokes. Bicycle wheels don't use aluminum spokes. If those spokes were that greyish color when they were new then I'm guessing they're cheap zinc-plated spokes which are not as strong as stainless steel. If they are, they'll be rusting where the plating has flaked off. If they turned grey over time, then it's probably a case of severe corrosion. This also weakens a spoke.
Finally, if the wheel was not properly tensioned as will be the case with many cheap machine-built wheels, it will be prone to breakage.
In short, I wouldn't go into the LBS with a particular solution in mind. I would go in with the wheel with the snapped spokes and ask them what the cause is and what you should do.