Mostly, I'll leave dwarves with their given names. However, if one does something to earn my ire or my favor, I will give them a nickname or profession name to reflect it. A dwarf that kills a beast may get the job "[Beast's name]'s Bane", a fortress guard member who murders a dwarf for violation of a mandate or other minor offense will get a title along the lines of "Police brutality poster child" or just "Brutal Pig" (after the squad name of every fortress guard in my forts, The Brutal Pigs), and vampires tend to get the job "leech" before they meet their long-overdue demise.
Particular favorites of mine, historically, have been the Baby Beastslayer, a child following its soldier mama around who managed to counterstrike and get the last hit on not one but two forgotten beasts; the Excelsior, an otherwise-uninteresting dwarf who managed to be the sole survivor of a forgotten beast rampage; the Seige Breaker, a new recruit who was the first soldier on the scene to meet a seige and managed to take out a goblin squad, a troll squad and a clown law giver on his own, ending the seige before anyone else could arrive; and, of course, V.L.A.D. (Vampiric Lever-Activated Defense), the bloodsucking leech who ran the traps in one particularly successful fortress of mine.