Is there a recall for the iPhone 3G? My iPhone 3G which is a 16GB version can upgrade to newer iOS in the past, but for merely 2GB of total usage on this iPhone, iTunes will take an hour or more to "Back up the data".

I know that some slow flash memory may take 2MB/second for data transfer, but even at that speed, 500 seconds should be good for 1GB of data, and 1,000 seconds will be good for 2GB of data, so that should be 16.67 minutes only.  (and, 2MB/s is usually a bottleneck speed for data writing, not for reading.  The speed for slow flash memory read that I usually see is 6MB/s)

Is it a design issue or might there be even a product recall for this issue?  If the iPhone 3G is taken to the Apple Store's Genius Bar, will they be able to swap another phone that takes a lot shorter time?  (My iPhone 4 and 4S, for example, is a lot faster for backing up the data when upgrading iOS).
The iPhone 3G is almost certainly out of warranty by now. You can't even really get a swap on it anymore in store. They've been end of lifed for quite some time and the store would have to order one for replacement. Since it's probably out of warranty, you would have to pay for it. I'm not aware of any warranty extensions for issues like this, they are usually done for obvious issues such as unexpected failure that's not due to regular wear and tear.

In general the 3G might sync a bit slower than an iPhone 4 or 4S, but it still probably shouldn't take 2 hours with that much data. My wild guess without seeing the phone is that the part where you're hanging up is the backup. You can verify where you're getting stuck by watching the progress indicator at the top of the iTunes window when it's hanging.

When I've seen this before, most of the time it wasn't a hardware issue with the phone itself. It could be something corrupt in your backup, or an app that has a lot of data stored in it or a setting or even just a large camera roll. It could be an issue syncing contacts or bookmarks. I've also had luck with disabling the setting that sends crash logs to Apple or resetting the sync history in iTunes.

Another thing to check is does it have a large amount of used space marked as "Other" on the bar graph (at the bottom of the summary tab) when it's plugged into iTunes? There should be some space taken up by this, but if it's a GB or more you might have something corrupt in the phone that's hanging it on sync.

It might help to sync it and watch what step it's hanging on and to post back with that information. That way, we can give you a more specific answer on how to fix it.