Making your own games or recreating others can be accomplished quite easily with an Arduino or compatible board, and even simpler contemporary versions such as "Flappy Bird" have been made quite well. One example of this is by hackaday.io member huy whose Arduino-based version mimics the original.
Huy has used a Nokia 5110-style LCD as shown below which has a satisfactory resolution for a playable game. The user interface is still a single button however you could add another to make control easier for your own version.
If you're looking for a neat display for an Arduino or Raspberry Pi, check out our 128x128 pixel OLED Module. With a diagonal size of 1.5" and 16,384 colours to select from, so almost anything is possible.
Furthermore there's a microSD card socket, and removable tabs on each side which can hold LEDs and buttons:
And using the module is made simple - we have tutorials and drivers for both the Arduino and Raspberry Pi platforms - great for experimenters or those who use both systems.
Furthermore, check out the forum where members are already creating modified drivers to rapidly increase the display speed. For more information including our Quickstart guides - and of course to order - visit the OLED Module product page.