Hello, all. Just catching up.
You’re providing the answer yourself:
‘Anything’ doesn’t mean literally anything.
What then does it mean?
There is a basic division you can begin with: good things, and bad things.
Not so fast, Tom. You're running far ahead of what is warranted. Let's go back to your example of your niece. Just because you can decide not to give her a dangerous, "bad" thing if she asked for it, that doesn't mean we're finished.
Let's take another look at it. Supposing she asked for a painting set. Would you give her a tennis racket instead?
Using those definitions, you can arrive at an understanding of what Jesus means by ‘good’, very generally this would mean things that are beneficial for the community of god. Comparing this idea also with how Jesus spoke about and demonstrated in actions his ideas regarding things like bodily comfort, wealth, things of that sort, those are generally not the kind of things that would necessarily be included, as a kind of heuristic.
Now you're adding another qualifier. And, basically, you're making it up. Jesus promised quite clearly: ask, and it will be given to you. Just because you and I have agreed, for now, that He must have meant "ask, and it will be given to you, unless it's something stupidly dangerous" that doesn't mean you have a license to interpret Jesus as saying "Ask for something, and I will decide what the best thing to give you is."
Basically, Jesus made a promise, and you are taking it upon yourself to open a loophole in it so large that it drains it of any meaning.
I'm glad you made an analogy of your niece asking you for a birthday present. It illustrates the gap between how rational people act and how God acts nicely. If your niece asked you for a book of fairy tales, you'd get her one for her birthday; and if you didn't, you know you'd be seen as doing wrong, as breaking a promise you made her. It wouldn't have made much sense if she'd said, "Uncle Tom, please give me a book of fairy tales," and you said, "Sure thing! Look, I got you an encyclopaedia! Happy birthday!"
Regarding ‘proof’ of answered prayers, it would be useful to define what higher authority or overarching notion of objective proof or über-standard you are referring to.
Cvanswey has dealt with this already.
Hello, cvanwey! Your help is most welcome.
A better question might be, if we truly think God answers petitionary prayer (ever), then we must ask ourselves, as @BigV mentioned many posts back...
1. Why has God never answered the call to prayer, in request for restoration in an amputee's limb(s)?
2. Why has God never accommodated the parent's request to have their child no longer have Down's syndrome? Just two examples, off the top of my head...
God promised that if we ask, it shall be given to use. But clearly, this is not so.
I think this is time to bring up that old quotation - from Emile Zola, I believe: "The road to Lourdes is littered with crutches, but not one wooden leg."
Well, clearly you feel that those questions must be answered, I don’t. What could possibly be a definitive answer to that, short of God coming down for a cup of tea and a chat?
Because, as cvanwey pointed out, God is not going to answer those questions Himself. Fortunately, He has you, and this is Christian Forums Apologetics section, a debating forum which exists for the purposes of Christians defending their faith with rational answers.
You don't have to answer cvanwey's questions and mine. Just say "sorry, I don't know the answer," and be on your way.
Because if you are here, you're here to defend your faith. With reasons and evidence, please.
Can we objectively distinguish 'answered prayer' apart from 'luck', 'chance', 'hard work paying off', other, etc..?
Or maybe, God never answers the call to request for Down's syndrome and amputees, but does for the ones you can equally attribute to 'luck', 'chance', 'hard work paying off', other, etc..?
Exactly. If God does answer prayers then, quite simply, it should be observable that Christians are "luckier". This should be a measurable phenomenon. Is it?
Objectively? Can you define what you would see as an objective standard? That could be worth discussing I think.
Nice try in deflection, but this is not what I mean

You understand the difference between subjective and objective, right?
Good point. Tom, please stop ducking the question.