Fortunately, being a halfling in 5e Forgotten Realms makes it easy to see how a halfling could be mean-tempered. After all: their homeland was just flooded, and many of them left homeless beggars in human slums. If she grew up on the streets, this explains why she's a rough-and-tumble fighter, and a dual wielder--a feature we associate with roguish personalities. I very nearly swapped her class to rogue, in fact, but ultimately decided I wanted the heartier flavour of a Champion fighter.
The only bit that still stood out was the violin. Even on her PF stat block, there is no mechanical representation of why she is so attached to the instrument, nor if she can even play. There is likewise no mention of an explanation in her introductory paragraph. I knew my players would be inclined to return Rosie's violin if it meant earning her friendship, and knew they wouldn't accept 'just because' as a reason for why the halfling possessed it in the first place. However, while I wanted an explanation for Rosie's musical ambitions, I also did not want to add an element to her character that did not do something to further the story. Thus, ultimately, I looked to the Factions of the Forgotten Realms setting.
The Inspiration
Gandalf! OK, not really Gandalf, but rather I decided to invoke an old story-telling trope involving a wise mentor, who usually dies early on in the adventurer's career. If Rosie's character was a rough-and-tumble street kid, and her music was the only hint of a softer side, then I decided it must come from someone who was dear to her, and who was taken from her prematurely. Perhaps someone who tried to get her off the streets and show her a better way of life. With the item in question being a string instrument, I couldn't resist using a Harper.
The Final Form
Ultimately, this is the background I gave Rosie on her character sheet:
"Short & fierce, Rosie was born in Luskan, where she learned the harsh realities of street life. She was a thug & a bully, & saw no need to ever aspire to be anything else. Then one night, she got in over her head. About to be pummelled, she was saved by the surprise intervention of a middle-aged gentleman who took out her assailants wielding only his reinforced bow string. The man was a bard named Gilian, a sailor who travelled the seas fighting & entertaining. He became a mentor to her, like the father she'd never had. It was Gilian who taught Rosie to love music, & that there could be more to life than fighting for survival. He told her stories that softened her heart, & made her realize a romantic side of herself. Rosie had planned to go with him on his next adventure & studying as a bard, as soon as he returned to port from a brief trip at sea. He didn't return, however, & Rosie later learned his ship was lost. Disheartened, she went on a wild spree of drinking & wasting away, which eventually landed her on the Wormwood."
I also gave her a Harper Pin as her only token of her mentor, which she wears as a necklace. I felt that this gave her a direction to grow in, should the PCs recruit her, and possible plot hooks there. I also purposefully avoided making the violin a possession of some lost love, as I wanted to use that background on another character. I also suspected that, if the PCs latched onto her for no other reason, the hint of Harpers might alone be enough to make them curious about the halfling.
I statted her as a Stout Halfling, and a Champion Fighter. I very nearly gave her the Urchin background, but ultimately settled on Entertainer, figuring her ability to play would probably be the prevailing plot hook if she stuck around long enough for the ship to make shore.
I gave her the personality traits of speaking her mind (loudly), and secretly being a hopeless romantic, both of which are implied in her PF description. For her ideal, I gave her a 'follow your dreams' to go with her newly-acquired back-story. For bonds, I gave her her violin, and for a flaw, I stuck with her big mouth.