Between that two I would probably get the S3 mini.
1. You have an official Android 4.x ROM. Officially, the DHD is stuck at Android 2.3.5.
2. The the 5 MP camera of the S3 mini is probably is as good as the 8 MP on the DHD. I have not tried the S3's camera, but the 5 MP on the Samsung built Galaxy Nexus does a better job overall than the 8 MP on the DHD, which can produce some great shots and some really bad ones.
3. Battery wise, the S3 mini has a larger battery and probably better battery life.
Me I love my DHD. It is well built and now runs a custom ROM and overclocked. If I were buying something now, I would probably opt for a more current model.
Actually I would look at the Galaxy S II. Might not cost all that much more and has Android 4.1.