An in-universe reason for the colors may be that each Istari's color reflect which ValaVala he was aligned with.
Saruman - Aulë
Gandalf - Manwë
Radagast - Yavanna
Alatar - Oromë
Pallando - Oromë
This theory is supported by the fact that both Alatar and Pallando (the two Blue WizardsBlue Wizards) are aligned with the same Vala.
Another reason (and my personal interpertaion when I read the book) could be that the colors had to do with rank (either official or symbolic). The three wizards featured in the Lord of the RingsLord of the Rings trilogy are Saruman the White, Gandalf the Grey, and Radagast the Brown. It could be argued that these colors be ranked based on their "purity" and therefore relative power. When Saruman declared himself as Saruman of Many Colors, he was claiming to have ascended beyond White which characterized his arrogance. Similarly, when EruEru intervened to restore Gandalf's life, Gandalf was "promoted" to White. Gandalf the White later claims that he is "Saruman as he should have been".
I suspect that The SilmarillionThe Silmarillion and Unfinished TalesUnfinished Tales may have more detail on this subject.