Bellatrix
Bellatrix Black Lestrange lived up to her name.
Black
Her original surname was the one she loved most. She was born a Black and remained a Black, even after marrying Rodolphus. For she was, indeed, Black in heart and mind—and her Black nature was irrepressible.
Bellatrix took extreme pride in her surname for it was exactly what she based her life upon. Her breath upon. Her own self upon. It was also why she hated Sirius so intensely. . . for Sirius scorned the name Black: he rejected it after being a Black himself. Being a Black was akin to honor but without the humility—for being a Black was a birthright. And Sirius spurned his name and what it stood for. And she hated him for it. And she expressed that hate by killing him for just that reason.
To be a Black was to do what was expected of oneself. But she was not like Narcissa, who did so in a cold, passive manner. No, Bellatrix was far more passionate in this obedience, for she was of a more willful nature than her easily submissive sister. Bellatrix had more options than Narcissa did because her will was strong enough to meet them. But the one she chose? Black. Her fervency in being a Black, in Toujours Pur, surpassed those of her parents. For her parents never killed, or tortured, or died for Toujours Pur. . .
Lestrange
Lestrange was not as special as Black. But she would never phrase this out loud, for it would be personally degrading, since she herself was now a Lestrange. For after all, her husband and her brother-in-law were much more tenacious in their beliefs, much more than the Death Eaters who renounced their connections to the Dark Lord and avoided Azkaban by doing so. She hated the lot of them. Hated them. It was terribly difficult to not fully express this hate, so she took her anger out on the Mudbloods and blood traitors instead. It was comforting, however, to know that the Lestranges hadn't been one of those traitors, and this knowledge proved to her that she had, indeed, made a good choice of her husband.
For contrary to what many thought, she had received some say in whom she married. Many proposals had poured into the Black household once she had graduated Hogwarts and thus her options as to what her future surname would be had been endless. Nott, Crabbe, Rosier, Yaxley, even Malfoy. But she chose Lestrange for she knew that Lestrange bore a kind of ardor greater than the rest. And that ardor was rather attractive. Much more so than Malfoy, the name chosen for her sister Narcissa. And—oh—Tonks? The very thought of Andromeda discarding her name—Black—for one like Tonks made her sick with disgust. Lestrange was a name worth exchanging, and so was Malfoy. But Tonks?
She hated her sister for commuting the name Black for a positively unworthy, detestable and revolting Muggle name like Tonks. She later expressed that hate by killing her niece, Nymphadora, who had preferred her father's name and had gone by it as if it were her first.
Bellatrix
In Pureblood society, not many people acknowledged a woman's first name as much as they did her surname or her husband's name.
Her mother, Druella Black née Rosier was known more commonly as the wife of Cygnus Black. Walburga Black was referenced to only because of her last name, a name that had never changed as she had married her second cousin, Orion Black. Lucretia Black was never known as an individual, but only as the daughter of Arcturus Black who married the still Pureblood but more liberal Ignatius Prewett. (And how she, therefore, became Lucretia Prewett.) Even her sister, Narcissa, was seen as a Malfoy above anything else.
Bellatrix was different.
She was called Bellatrix Black and Bellatrix Lestrange since both names were accurate to her being. Both Black and Lestrange hinted at her aristocratic Pureblood status, but when others uttered it, sometimes out of fear and sometimes out of hate, what mattered more than Black and Lestrange was the first half of both her names.
Bellatrix.
She was known as Bellatrix and not Black, Bellatrix and not Lestrange. Bellatrix the murderer, Bellatrix the merciless torturer, Bellatrix, the Dark Lord's insane follower. . .
The name was of Latin-Greek origin, as most of the names in the Black family were. It was also the name of a star, as, yet again, most names in the Black family were. Bellatrix marked the left shoulder of Orion and was the twenty-seventh brightest star in the nighttime sky. Brighter than her was Sirius, a fact that she did not usually pay heed to but secretly annoyed her nevertheless.
There was no particular story behind Bellatrix as there was Andromeda and Narcissa, but Bellatrix stood out on its own.
For Bellatrix meant female warrior.
Indeed, Bellatrix Black Lestrange did live up to her name. She killed for it too. And in the end, she died for it.