could/might

[licinio]

Can I use "could" instead of "might" in the following sentences?

(suggestion) you might try phoning directory enquiries
(annoyance) you might at least say you're sorry

Thanks.

[5jj]

Yes

[licinio]

My son asked if he could give me back what he owned the following week.

Is MIGHT possible here in replacement of COULD?

Thanks.

[5jj]

My son asked if he could give me back what he owned the following week.Thanks.

'Might is possible', but 'could' is far more likely these days.

[Raymott]

My son asked if he could give me back what he owned (owed?)the following week.

Is MIGHT possible here in replacement of COULD?

Thanks.

Yes it is. 'Could' is more common, at least where I come from.

[Mohammadhelmi]

My son asked if he could give me back what he owned the following week.Thanks.
both of them are acceptable answers
but the difference resides in the degree of certainty in your example.
might shows more degree of certainty than could

[5jj]

b Both of them are acceptable answers, but the difference resides in the degree of certainty in your example.
might shows more degree of certainty than could

Not it doesn't.

The original question for could was "Can I...?". Though this could just about suggest a question about the speaker's ability/possibilty, it almost certainly requests permission.

The original quetion for might was " May I ...?" This requests permission.

[Raymott]

might shows more degree of certainty than could

This is nonsense. It's not your fault; you were taught it. But I hope you don't teach it and perpetuate the myth.
In general, you cannot rank the modal verbs in terms of probability.