|Most constellations are a Latin word that translates to some generic thing. But a few are their own translation such as a specific person's name. The Phoenix is a unique mythical bird that is pretty much only known as Phoenix, the bird that's hard to get rid of. As we all learned in mythology class, or from Harry Potter, when the bird dies it burns in a flash to ash, and then the ash turns back into the bird and it starts all over squawking and flapping and pooping on a newspaper.
The idea of a phoenix, "burn, reanimate, repeat" is an ancient concept and very cross-cultural. Some of these ideas were shared early and widely and seemed to catch on for some reason. A bird that burns itself back to life, it just seems so natural.
There is a grandpa star here, HE0107-5240 which is one of the most metal-poor stars known in the Milky Way. Remember, astronomers have a kooky idea of what a "metal" is. To astronomers, there are only three things in the Universe; hydrogen, helium, and metals. That drives the chemists crazy because they went to so much trouble to make the periodic table look like a game of Tetris. So anyway, this star has only 1/200,000th of the elements bigger than helium when compared to the Sun. This is probably because the star was born before metals were invented.
There is one meteor shower associated with the constellation, the Phoenicids, which occurs around December 5. It's caused by a comet that burned, died, and was reborn as a bunch of bits of gravel that orbit the Sun.
Robert's Quartet is a tight little group of galaxies in Phoenix. The biggest members are rushing into an intersection and all ignoring the four-way stop signs. They are rather dim so don't expect to find it with that refractor you bought at a yard sale for five bucks. That one is for smacking people who get nose prints on the lens of your real telescope.