Choosing components to clone ZX80
The Z84C00xxx is not exactly 'modern', but it is still available new. Another advantage is that being CMOS it uses much less power than the NMOS version. It is specified to work at TTL logic levels, so for best compatibility I would use HCT logic chips.    

For SRAM I would use a 62256 (32kx8) because they are readily available and often cheaper than smaller capacity static RAMs. If you can't get it in DIP then use a SOIC to DIP adapter board.

For ROM you can use either EPROM or (parallel) EEPROM. I prefer EEPROMs because they are quicker to program and don't need a UV lamp for erasing. The AT28C64 (8kx8) is readily available in DIP package at a good price.