Are you talking about making an actual Game&Watch game on game&watch type hardware, or just making a computer program/game that looks like a Game&Watch game?

Here's an interview with the original team behind the G&W, it has some details on the hardware used and how it works: http://www.plasticpals.com/?p=23596

It's a very simple hardware with a custom made LCD screen, using the same type of simple cpu an 1980:s desktop calculator would.

Making the custom LCD is probably the hard part today, for CPU you could use any small modern microcontroller, like a pic, AVR or a tiny arduino, the processing needed to drive the game is very little. It basically just lights up LCD segments one after the other, increasing speed to increase difficulty.

Since it needs a custom LCD, and the program is in ROM, it's not really possible to modify an existing game to something else.

I see the instructables guy used a general purpose LCD instead with actual pixels. This is probably how you have to do it today. Not really game&watch, but I guess it feels a bit like it :)

For prototyping, using Arduino is probably with the least steep learning curve. You can program it in "processing" (a c-like language made for rapid prototyping and to be easy to learn) or in C and C++ (or assembler if you want)

For mass production, it depends a lot of how big mass you want.

An arduino is really a protoyping platform made with either an AVR processor, or a tiny ARM processor. (the first one is very cheap and not very powerful, the other also cheap but a bit more expensive, and a lot more powerful).

I'm guessing an AVR would be plenty for a g&w with custom LCD, but if you want a bunch of pixels, the ARM version might be better.

You can prototype and learn about hw dev on Arduino, then when you are ready for mass production, switch to using a custom AVR/ARM hardware which is similar to an Arduino but with all the prototyping extras removed.