Thanks very much for the extra questions folks, that's been very helpful. Sorry in advance for the big post, but this should make things a little clearer.
The premise of this is a football board game, which involves forward passing, like gridiron. All these LEDs are in order to allow the board to beautifully illustrate whether a pass is a quick pass, short pass, long pass or a really long pass (long bomb).
Does the lighting have to be part of the switch?
No it doesn't - but the lighting was going to be dependant upon the switching. I am trying to build a real world example of this little web app:
http://www.bloodbowlonline.com/InterceptionApplet.shtml however with a lot more features.
Screenshot (of the web app):

However rather than text to tell you whether a range is quick, short, long or long bomb, I would like the LEDS to light up with different colours like this:
I coded up the logic in excel to make sure that I could figure out lighting up the correct cells.
How big are the squares? EDIT: 1" I just saw that.
Is pressing OK or do you need just touching?
I don't mind, however the surface of the squares will also be taken up by gaming tokens that don't fill the whole square inch. At this stage I am open minded as to how to engage the arduino to say - "the starting spot is here".
Does the project have to last forever or just get through the weekend?
I'd really like this to last. Funnily enough, I just wanted to build this as a game board initially, however since I have been looking around and seeing the very cool things that you can do with LEDS, I would also like to use this board to be able to do some of the cool things, such as play checker's, ony without using any game pieces, or just some funky light work.
Does it matter or people press > 1 at a time?
For the game board, the order of the pressing will matter. The idea is to track how far a football can be thrown, so it needs to start somewhere and end up somewhere. I am happy to have a "Start" button to the side of the board that needs to be pressed to allow the LEDS to be engaged if that makes it simpler to avoid confusing the arduino.
Is it a requirement that people press > 1 at a time?
No, that is not a requirement.
How far apart do you have to space these things?
The board itself will need to be close together, so each 1 inch square will need to but up to the next one - like the images above.
Can they be on one board or do they need to be separate?
They can be separate, however for game time, they will need to butt up against each other as above. That said, I am happy to build this in multiple parts (halves or quarters) and bring them all together to play the game - as long as there is a way to tie all the elements together so that the calculations work out ok.
So basically, as the game plays out, if you want to pass the ball I had originally envisioned pressing a "Pass" button at the side of the board, then pressing the switch on the source square and it would light up the squares with the appropriate colour to indicate the range. You would then press the destination square, and it will then light up the path between the two squares to indicate that the ball could be intercepted along that path (as per the web app screen grab).
I initally thought of an mdf board with an LED and a switch sticking through for each square. Then I thought that maybe a captive clear plastic 1 inch square could shield the led and diffuse the light, with the switch poking through. My next thought was making every square into a clear topped switching mechanism with the led below it, thus protecting the switch itself and the led from any accidental contact.
The really cool thing about this idea is that I have been talking to my kids about it, and they are keen to lend a hand too ! So this will be a nice project for the whole family (although Mum isn't that interested !).
As for budget, I am hoping to keep the cost down, however I understand that this many LEDs alone will cost me about $100. I'd like to keep the switch element cost down too - I saw an all in once LED lit switch today with 2 LEDS inside it, however the total cost for 390 of them would be about $3,500 - that's significantly out of my budget. That said, for the number of I/O pins, I think the arduino mega will be the way to go, and I think it's cheap for what it can do.
Thanks again for your assistance !