The phrase "have a good rest of the week" is correct and can be used in written English. You can use it at the end of a conversation with someone, as a polite way to say goodbye and wish them a good week. For example, "It was nice talking to you. Have a good rest of the week!".
I hope everyone has a good rest of the week.
Head to your next class, and have a good rest of the day.
Have a good rest of your week".
"I had a good rest during the off-week last week," Choi said.
Have a good rest the night before.
Have a very good rest-of-weekend.
Even though you have a good introduction, the rest of your story could stink.
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