As @RedDeerGuy1 said, it may have been a peacock (firebird in Russian folklore is very similar to phoenix, and is sometimes depicted very similarly to a peacock). Some others described it as an eagle, or a rooster. Wiki page on it states the looks of a phoenix vary depending on the writer. New World Encyclopedia offers various inspirations for the origin of the bird in different cultures.
In other words, it’s not definitely known what exact bird inspired the myth. All that is constant in all variations of the myth is that phoenix is a big bird with many colourful feathers and ability to be reborn.