When someone says "he has a moment" or "he has his own moment," what does that mean???

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"he has a moment" sometimes means that the person has time available for a meeting or discussion.

Examples:
"He has a moment now if you would like to speak to him."
"Can you ask him to come to my office when he has a moment?"

In many cases though, if someone "has a moment" or "has his own moment", it refers to period(s) where the person acts in unusual or atypical ways (different than normal), or moment(s) when the person experiences a certain emotion (e.g. fear, anger, etc.) or sometimes when someone just takes a moment to think about something by themselves.

Examples:
"He had a moment of shock when he realized his wallet was missing from his pocket."
- - -
"He had his own moment of contemplation to decide whether to go to the concert or respect the 9:00 PM curfew his mother had set."
- - -
A: "Your child is so well behaved."
B: "Well, he has his moments" (sometimes he acts well behaved, but often he does not)

@hirofumi usually you would say "he has his moments" in that case.

that would be "he has his moments"

like the previous example:
A: "Your child is so well behaved."
B: "Well, he has his moments" (sometimes he acts well behaved, but often he does not)