By the time J.K. Rowling published Harry Potter and the Order of the Phoenix in June 2003, she was on the fifth Harry Potter book. So by then, we all knew the drill: it would be called Harry Potter and the Something or Other. The important part of the title is that magical “Harry Potter” up front. That’s what brought out the crowds to buy the novel's first run of 8.5 million copies – more than any other first edition at the time (source: Deirdre Donahue and Jacqueline Blais. "Tall 'Order' for Mr. Potter." USA Today. June 5, 2003).
Still, even if "Harry Potter" is the important part of the title, "Order of the Phoenix" has a nice ring to it. Harry Potter and the Goblet of Fire introduced us to the secret society of the Dark Lord's followers, Voldemort's Death Eaters. Now, we have another group of people organized around the Phoenix. There's only one man in the series who has a phoenix: Albus Dumbledore, whose phoenix Fawkes saves Harry's life with his healing tears at the end of Book 2. So, "Order of the Phoenix" suggests followers of Dumbledore – a good group of people to face down the evil Death Eaters. For the specific symbolic significance of the phoenix, check out "Symbolism, Imagery, Allegory."