In Naruto Part 1, Naruto's headband was blue. In Naruto Shippuden, his headband was black. In one of the movies, it was red. A shinobi's headband symbolized who you are loyal to, so is there any significance to the headband's color, and if not, why was it repeatedly changed?
As Kuwaly state, there does not appear to be any significance in the color of their Headband's strap. The most common of all colors are either the blue or black
Navy Blue: The most common colour of forehead protectors. Many ninja keep this colour for the entirety of their ninja career.
Black: The second most common colour of forehead protectors. Some, such as Naruto Uzumaki, even switch from blue to black after the timeskip
Now the reason to why they change colors can be due to either
- They do need to be replaced from time to time, especially after difficult training
- They simply out grown of them, and want to change the color
In Shippuden episode 6, when Naruto is talking with Iruka over ramen, Naruto says to Iruka
The cloth on the precious Headband you gave me got all tattered
So given that information, Naruto changed his headband due to the fact that it was simply torn and needed to be replaced
Based on this Wikia page, there isn't any particular significance to the colors, though some of the variation is based on location/loyalty. Navy and black seem to just both be common colors. Other characters switched color after the timeskip, such as Sakura and Choji, both of whom switched to red.
Theories on why various characters' cloth color changed include that it's to match their new look or that it's based on rank, but there doesn't seem to have been anything specifically said by the creator (that I can find), and neither of those theories are particularly consistent with the rest of the show. It seems more likely that the changes were just done as a part of the stylistic/look changes post-timeskip.