Not all builds of OS X v10.7 are alike. What originally shipped via the Mac App Store was build 11A511, but what your Mac mini shipped with—and requires as a minimum version—was build 11A2061. It would be interesting to know which version of 10.7 you downloaded from the Mac App Store. If it was 10.7[.0] 11A511, then your results are no surprise. If it was 10.7.1 or anything later, then I'm surprised you couldn't boot from it. If you really did download it exactly 6 months ago, you should have 10.7.1. But if you downloaded it in July or early August, you probably have 10.7 11A511.
By the way, what exactly did you burn to the DVD? Did you use the InstallESD.dmg from inside the "Install Mac OS X Lion" app bundle?
If it turns out that you only have the original 10.7 11A511, you won't be able to get the Mac mini to boot from it. If you have another Mac available to you, you might see if you can find a way to re-download Lion from the Mac App Store, because what you'd download now would be 10.7.3.
If you just want to get Lion reinstalled and don't specifically require doing it from DVD, then please note that your Mac mini is probably new enough to have EFI firmware that supports Apple's Internet-based recovery option. Just hook up your Mac mini to a broadband Internet connection, and hold down Command+R at boot and it will attempt recovery from the recovery partition on the local hard drive if there is one, and switch to Internet-based recovery otherwise. If you don't have an Ethernet connection, don't worry, there's a way to have it join a Wi-Fi network as long as your network doesn't require enterprise-grade authentication.