Nickelodeon's 2005 series, master her birthright as a firebender and perfect the technique of emitting blue fire is immaculate, but why her specifically?
With the introduction of her arrival at the end of the first season and within the second season, Azula's demeanor as the next intimidating threat for Team Avatar to encounter had the audience understand that this is someone not to be taken lightly. As this formidable foe, Azula's power and authority were unmatched every time she was on-screen. The fact that she's powerful enough within the show to even possess the ability to wield blue fire, but control it to her whim is a key reason she stands out as the only firebender who can utilize it.
One notion many believe in is that her proficiency in the "cold-blooded fire", also known as lightning generation, caused Azula's main source of fire to turn "cold" or blue as a result. This doesn't entirely fit, though, as in the sequel series, The Legend of Korra, Mako could generate lightning from his fingertips and still use normal red or yellow fire. Throughout the series, it is suggested that the creators based her strength and control over the blue fire through the simple scientific principle of fire. A blue flame has been proven to occur at a higher temperature than yellow or red fire and become complete combustion. However, this isn't the only reason Azula may be able to wield blue fire in the show.
Throughout her life, Azula was praised as a natural-born prodigy by her family. Her power and ability in firebending caught the eye of her father especially. When she was 9-years-old, Azula demonstrated and even flaunted her firebending prowess to her grandfather, the late Firelord Azulon, even surprising him how great she is for her age. She grew to love this constant attention and become somewhat of an exhibitionist overtime as she would typically demonstrate her power and sadistic nature over her peers to garner their attention too. After her brother was banished from the capital to look for Aang, Azula spent the next three years relentlessly training her bending, strategy, and combat until she concentrated her skills down into her refined, advanced blue fire technique.
One final notion may be based on the creators' having personified Azula on color symbolism alone. While her namesake in the Avatar universe derived from her grandfather, Azulon, but her name also comes from the Spanish translation of blue, which is Azul. As she is the only one who is personified as blue amongst a nation of red, this makes her stand out even more as a character. Moving forward with that concept, the creators could have structured her as a sort of contrast to her nation, especially her brother, Zuko.
While throughout the series, Zuko had been very ionate, hotheaded, and impulsive, Azula had been depicted very even-tempered, cunning, logical, and a perfectionist. However, someone who personifies the color blue showcases confident and self-controlled, but under the surface, may be hiding vulnerabilities. Although there are many possibilities for why Azula was gifted with blue flames as her signature technique, her upbringing as someone who must always strive for perfection and attain a flawed ideology of world domination is one major reason she was able to achieve her iconic ability in Avatar: The Last Airbender.