With you stub configuration, your stub router sends info on connected and summary routes upstream. You can minimize what it sends using "receive only" (make sure you have static routes upstream if you do this).
In your case, you're trying to minimize updates received by the stub router. You may want to look into Address summarization. Make sure your upstream router is summarizing routes to you (if possible).
i.e. a summary of 192.168.16.0 255.255.240.0
includes networks 16-31. If network 17.0 goes down at your main site, you won't hear about it because you're getting a summary (specific routes get suppressed).
You can always use static routes vs. eigrp depending on the overhead (# of networks)