What are the best Chinese terms for the correct sense of the English adjective "everyday"?
“日常” is the closest in most contexts. It conveys the sense of "commonplace, mundane" very well.

Some example phrases:
everyday life - 日常生活
everyday conversation - 日常对话/日常会话
everyday clothes - 日常服装

It can also be used as a noun: 我厌倦了重复的日常 (I grew tired of my repetitive everyday life)