Yes, a good cry is somewhat idiomatic. The phrase has been around for quite some time, too. An 1869 magazine article read:
Completely overcome by chagrin and disappointment, I had a good cry over it.
and in an 1887 issue of Harper's:
How often in her own room, distracted between those conflicting fears and hopes, she sat down with her face between her hands and had a good cry over it!
More recently, in a 2004 novel by Steve Tesich:
He had a good cry over it and it made him feel better. Made him feel good again. Good and heartbroken. Good and scared. But basically good. He was feeling good about something again.
So, what is a good cry? The website AgingCare.com offers a hint:
Researchers have proven what many caregivers have already figured out on their own: sometimes there's nothing like a good cry to make you feel better. Crying is cathartic.
We can cry about a lot of things, but a "good cry" will make us feel somewhat rejuvenated and emotionally cleansed. When comparing the improved moods of criers as compared to their more "stoic" peers, one researcher said:
This pattern is often found in retrospective studies where people are asked to rate their mood levels after having experienced a good cry.
Going back to your original sentence, it's hard to say for sure why the person cried – not without knowing what was in the letter. Whatever emotion triggered the tears, though, we know it was a good cry, meaning that the cry was deep, and that some emotions were released to the point where the person felt "cleansed" afterward.
good of you to...
Your actions in a particular situation were appropriate, and maybe exceeded what could reasonably be expected of you.
It was good of you to take Geoff home last night: he was so drunk that I was worried about him!
good for you to...
Your actions are/will be beneficial to you.
It's good for you to get some healthy exercise every day
It is often used in a conditional form:
It would be good for you to get some healthy exercise every day
good to you to...
As this NGRAM shows, this expression is uncommon and, when used in this context, seems to mean the same as the for version.
It occurs more often in sentences like this:
Does it sound good to you to go to the beach tomorow?
Best Answer
It’s no good by itself means, approximately, “This is a bad situation and something must be done.”
It derives, however, from a longer expression, It’s no good Xing …, where Xing is a gerund phrase:
A variant of the expression is It’s no use Xing, and either version means “Xing will not make the situation better.”
Observe that “It was no good” is followed immediately by “He’d have to say something to her”. Mr. Dursley is telling himself that the situation has gotten out of hand and consequently he and his wife must end their long practice of not mentioning her sister’s family, because that practice is “no good”: of no use.