Hi there! This is such a good question and one I am MORE than happy to answer. Although there are many ways to find lore-appropriate names for the elf or "mer" races I'll try and keep it short and sweet. Some of those that I use most often are looking up previously used names throughout past games, finding the root of their language or naming system (Ie if the names are based in old french names I might look at old french names or words and use those and or combine them in a way that mirrors the ones used in-game), and finally looking at the known words for their lore appropriate languages and use those.
So let's say we are looking for a Bosmer/ wood elf name. I would search "
bosmer names." There I can see the names given throughout the various games and get a general "vibe" or sense of what the naming conventions might be. For specific games, they even mention the prefixes and suffixes used, so if I wanted I could just pick two I think go well together and use those or use an existing name. Ie Phy + Thia (as seen for female bosmer in Arena and Daggerfall) to make Phythia. I can see on this page that some also use surnames of a sort such as Ivyshade, so our character becomes Phythia Ivyshade. That is a very quick and simple way to pick a name that is fitting. Lets say I'm happy with Ivyshade but not quite happy with Phythia. From there I might look up words in their language or in ones that had some impact.
Example a) Example b) There I might find I like the word Liette (liettei) meaning Spriggan (an. n.). I could then make her name Liette Ivyshade or combine the word with a suffix or another name in order to get Liettethia or Lietteriel. Finally leaving me with Lietteriel Ivyshade. A name I am personally quite happy with, and might just use in the near future...
Hopefully, that makes some sense. Honestly just have fun with it! If you have any other questions or want help with generating some names my direct messages are always open.
Cheers,
Hlif