Try to avoid colors that fall within the original's degree range. For example, if original color works out to hsl(110, 98%, 50%), a yellow-green, should not pick another value between 90 and 120, which is the range for yellow-greens.
https://drafts.csswg.org/css-color-4/#hsl-examples
Randomizing saturation and lightness values will create more variation in random palettes. But may add clashing colors to the mix.