Let members pick their own name color. Offer a preset palette (organized into a single-select menu, auto-split across menus if it's large), or enable custom colors so members choose any hex color via a button. (Configure → Color roles for the palette; Custom colors for the pick-your-own option.)
A color that follows the member. After setting their own color, a member can press
Remember this color to save it on their Packet profile. In another server with custom
colors on, Packet then offers it to them ("Use #FF8800?") when they have no color there, or
when they run /color set without a color.
Nothing changes until they press Use, and your server's rules still apply: the required
role, the slot limit and the wait between changes. Members forget it with /me clear.