Lets say there's a new cone that detects ultraviolet light, represented here with U. Really, I think this would work with any frequency, I'm just picking ultraviolet b/c it's outside the range we can currently see, so its easier to understand it as a new colour. But in reality, it might have difficulty making it down to the retina, based on some things I was reading, so don't get too caught up on the U, it's a thought experiment.
Then we have these cones: R G B U
This gives us 8 new colours that we cannot currently see!
These are the combinations they could fire in, asterisk means that this colour is new, we've never seen it before:
R | G | B | U | Description
--|---|---|---|----------------------------------------
| | | | Black
| | | X | *unknown, we've never seen this colour
| | X | | Blue
| | X | X | *unknown, we've never seen this colour
| X | | | Green
| X | | X | *unknown, we've never seen this colour
| X | X | | Cyan
| X | X | X | *unknown, we've never seen this colour
X | | | | Red
X | | | X | *unknown, we've never seen this colour
X | | X | | Magenta
X | | X | X | *unknown, we've never seen this colour
X | X | | | Yellow
X | X | | X | *unknown, we've never seen this colour
X | X | X | | The old white, now unknown
X | X | X | X | *The new white
Note that it's also possible that our brains would reinterpret how we see the old colours, ie in the same way that what we currently see as white wouldn't make sense as white anymore, and would thus look different, we might see other colours as different, too!