Using 4 motors at a combination of full forwards, full reverse or dead stop gives 10 basic types of motion, with even more precise net movement using variable speeds. Above is a simple diagram.
My belief is that TLG (The Lego Group) should manufacture these parts as part of an expansion set for the, soon to be released (from the perspective of now), EV3 Mindstorms kit which, in an improvement over the NXT, can support 4 motors per control brick.
This would be a great way to also introduce a pack of bonus beams, gears and connectors and an expanded instruction book (or CD) for the EV3. This is entirely up to TLG upon 10k supporters as I have no particular interest in drawing up a set inventory.
In addition, TLG supports a robotics event, the First Lego League
and having talked with past competitors, many drive systems have significant physical limitations. A mecanum drive would increase manoeuvrability at the cost of programming difficulty.
Having looked at various mecanum designs on the internet, I have come up with a fairly compact model that would require three new moulds with two of the new moulds being useful in other, less specific, situations.
Any suggestions on how to improve the part designs would be appreciated. I am not particularly keen on designing more custom parts for this project, but should any ideas stand out, I may try and add them.
-Edit: I am working on an end supported roller design to minimise axle warping.
Note; I have not proposed an omni wheel because mecanum wheels are extremely expensive and there are manufacturer(s) that already make dedicated Lego compatible omni wheels at a comparatively low price. Please support anyway as you could always use the hubs here to make your own omni wheels.