There are several synonyms for the phrase "got some shut eye". Some alternatives to this colloquialism could include "slept", "dozed off", "rested", "napped", or "caught some Z's". Each of these phrases implies a period of rest or sleep, but with slightly different connotations. "Slept" is a straightforward substitute, while "dozed off" suggests a shorter or more accidental period of sleep. "Rested" might be used to describe a more intentional and peaceful period of rest, while "napped" suggests a brief and rejuvenating break. "Caught some Z's" is a fun and informal way to describe getting some sleep that can be used in casual conversation.