Meaning 'sometime during next week' or 'during next week/throughout next week'; exact nuance (sometime vs. throughout) depends on context.
来週中に見積書をお送りします。
I will send the estimate sometime next week.
来週中は会議が続くので、対応が遅れるかもしれません。
I have meetings throughout next week, so my response may be delayed.
その仕事は来週中に終わらせてください。
Please finish that work by sometime next week.