By default, all text in a ship name is white. However, while messing around with my ship's name, I learned that it is possible to change the color of a letter or a whole word/phrase. Before the text that you want to be changed, type out "^" [shift + 6] and then a number from 0-9. This will change the color of the following text, and also get rid of the "^" and number in the name. You can use different colors for different letters by typing "^" and another number once again.

^0 = black.
^1 = red.
^2 = green.
^3 = yellow.
^4 = blue.
^5 = cyan.
^6 = magenta
^7 = white.
^8 = primary faction color.
^9 = secondary faction color.

Say you want your ship to be named "pie" but want the p to be blue, the i to be white, and the e to be green. You would type the following:
^4p^7i^2e
and your ship name would look like this:

Thank you for reading this guide. If you have any questions, comments, or concerns, feel free to leave them below this guide; I will not delete anything. Please consider giving this a thumbs up and favoriting it, as both of those things feed my ego so much.

Credit to Xcyiterr for informing me about the nature of ^8 and ^9, as well as providing demonstrative screenshots.