RealmNameForgeBegin with a playable character idea and two choices you would be happy to hear in voice chat. A good shortlist includes a favorite and several backups with genuinely different second names.
Start with the character you will play
Pick the race, then choose a detail worth naming. A Dwarf who repairs tools suggests a different vocabulary from a Dwarf who patrols a pass. Write down one object, place, or habit to keep the search focused.
Compare names as they will be used
Read each complete pair aloud, then say its first part on its own. A good pair should survive a quick introduction. If the two halves blur together, swap one ending or opening sound and listen again.
Keep backups that solve different problems
Save the best pair as First Choice. Keep one Backup that preserves the first part and another that changes it. If the client rejects the whole combination, you can try the related pair quickly. If the problem seems to follow one half, try the other backup.
Check the finalist in character creation
Use your chosen region and ruleset in the Forever client. Regional uniqueness applies to the complete pair, and Blizzard can evaluate the two parts together. A local formatting check can identify issues to investigate before you get there.