There are two cases: a TV based games or a portable games console. You could do carriers for both of these cases.
A portable one could have an LCD screen and could be similar to the one that was done for the TinyFPGA BX:
For a TV-based one, HDMI connector would be used. It would need a headphone jack for audio. HDMI audio would be good, but I suspect that is too difficult.
It would then need a way to connect controllers. A Nintendo i2c connector as used by Wii Nunchuks and other Nintendo controllers. You can get wireless controllers that work with them. There is also an earlier form of Nintendo connector used on the NES, which again you can get wireless controllers for. (Although the one I bough to work with a NES classic did not work).
Or there could be bluetooth connectivity via an ESP32 or some other device, to allow Bluetooth controllers. Wifi is another possibility: ESP32 would provide both.
Another option is USB host, either driven directly from the FPGA or using an SPI-based device. This could potentially support controllers such as Xbox or Playstation controllers, or keyboards and mice.
PS/2 connectors for keyboards and mice are also a possibility.
A further possibility is to have a cartridge interface for playing games cartridges. For example a gameboy or an SNES cartridge interface.
And after that, A Virtual reality headset carrier would be good ...