I'm a professional teacher from eCineseLearning. (http://www.echineselearning.com) This phrase is a idiomatic expression, and hold two meanings: 1>To acknowledge somebody;2> to give somebody any respect or attention. Usually, it is used in the negative way, "not give (somebody/something) the time of day" For example, 1) If a worship doesn't think you are rich enough, he won't even give you the time of day. 2) After tthe couple were apart, they never give each other the time of day because their marriage life was not plesant.

This idiom usually comes in a negative form, "not give (someone) time of the day. This means that a person is refusing to talk to someone or spend a little time with someone because he think the other person is unimportant or they once had an argument. Examples: The popular kids won't give me the time of the day. They do not think I'm cool. She won't give me the time of the day. We had a huge fight yesterday.