I still think Uriel is trying to redeem Anduriel. I mean they share part of the same name so I think there is a closer connection between these two angels than say two other angels with different names.
Uriel apparently means something to the effect of "God is my light".
According to this reddit (
https://www.reddit.com/r/dresdenfiles/comments/3hs0bj/anduriels_name/), Anduriel is a made up name.
The "el" suffix, as we know, is the "of God" portion of an angel's name, so removing that leaves "Uri" and "Anduri", or possibly "Ur" and "Andur", depending on how the construction works. "Anduri" could be construed as a direct negative of "Uri", like adding "dis" or "un" to a Latin root. While Uriel would mean "God is my light", Anduriel could mean something like "God is my darkness". That would imply that when Anduriel rebelled, he chose a rebellious name, overriding whatever his original God-given name was.
Based on that reasoning, either Anduriel is a moniker chosen out of spite and as foil to Uriel's office, or it is the original name of the Fallen Angel with a different and unknown root. In either case, though, Anduriel's name doesn't seem directly related to Uriel's.
I could easily be wrong, though. Etymology excites me, and I will readily follow linguistic rabbits down numerous holes.