Just to update... this turned into a couple issues...
1) When I got home, I couldn't remote into the web server (I use Remote Desktop from another computer on the same network). The web server WAS working (I could access the pages on it), but I couldn't get into the box to do any maintenance. So I rebooted the box.
2) After I rebooted, I found the IP Updater program from Dyndns hadn't accurately changed IPs early in the morning (which I knew). BUT, I also found out the box had done an automatic Windows update at some point early in the morning and rebooted on its own. I then disabled the automatic windows updates. I'm guessing there's a connection between the automatic windows update and the IP update program not running correctly.
On someone else's suggestion, I loaded TeamViewer on the web server computer. I can now connect to the web server from my phone or my Kindle Fire. That doesn't rely on the IP being updated, so I should have a "safety" way in to the box.
And for what it's worth, my home IP address is changing as often as every 20 minutes, which makes no sense to me. Yes, a static IP would be the way to go, but $15/month is ridiculous IMO. $15/year I'd be willing to pay. But again, I'm a cheap bastard.