The short answer is that you should use "to".
The most precious thing to me
A better way to phrase it, although slightly more formal, would be:
The thing most precious to me
However in your sentence you are saying this "thing" is both significant and precious to you. The preposition has to suit both adjectives!
Fortunately "to" is also appropriate for "significant".
"For" in this context would carry the meaning of something relative to the speaker, so it isn't really appropriate here because the precious thing is not making itself precious for you. You regard it as precious, and so it is precious to you.
My explanation of the difference between to and for in this context has caused some discussion and so I will add further explanation.
you don't think that some of the examples for IN RELATION TO are similar to this sentence? For example "The ice-cream was a little bit [too] sweet for me"
But consider this very similar sentence:
"The ice cream tasted a little bit [too] sweet to me".
Similar sentences, one uses "to", the other uses "for". Why? The difference is that "taste" is relative. We all taste things differently, including degrees of sweetness. However, something either is, or is not sweet, so that is not relative.
Saying "The ice cream tasted a little bit too sweet to me" means that to your taste, the ice-cream is too sweet - but not necessarily for your taste. You might like it that sweet! But to you it tasted that it might be too sweet, by whatever measuring stick there is for sweetness.
However, saying "The ice-cream was a little bit too sweet for me" brings your personal taste into it. You are not commenting on how sweet it may be, or how others may taste it. It is too sweet for you. In fact, this version of the sentence does not even address the literal "taste" of your tongue - you could be saying that it is too sweet for you for some other reason! Maybe you're on a diet, or diabetic, and just can't have too much sugar.
The meaning of these two sentence are very different. The sentence with "to" focuses on how it tasted to you, but makes no direct comment on your personal feelings. The second, using "for" addresses the sweetness of the ice-cream in relation to you.
@JavaLatte went on to state:
Personally, I don't see any difference at all between precious and sweet. They are both impersonal (precious metals, sweet oranges) unless you make them personal by adding "to/for me"
But there is a difference. Just as "taste" can mean your personal preference (which is relative) and also the literal taste on your tongue (which is not relative - remember something is sweet, or not), similarly the word "precious" has more than one meaning. @JavaLatte says it means something is valuable (not relative - things like gold, silver, tin etc either are precious, or they are not). But it can also mean something has sentimental value, and that is entirely relative! Something of sentimental value to you may be entirely worthless to everybody else.
Which returns us to the OP's original sentence:
"In spite of the popularity of modern international music, the traditional music still the more significant and precious thing to
Music has no intrinsic value as such. And certainly any "significance" (which the OP also uses along with "precious") is personal. It is clear the OP is talking about music which is personally valuable to them, and I have no doubt at all the correct expression in this instance is "precious to me".