encapsulating a positive thing among many negative things

Irrational exuberance would seem to fit except that the term has been used to mean something (only slightly) different. No doubt the beneficiaries of DotCom believed that they had found something very beautiful. On a slightly different note, I think that an aspect of your question that makes it difficult to answer is that it is not clear whether, from the point of view of the article, the capacity to take the negative and see it as beautiful is a desirable thing or whether it is portrayed as psychoticism.

I like the phrase how

a flower emerges from a pot of dirt

Not sure whether it's applicable or not, but when I read your post this came to mind.

Perhaps you could adapt it to suit the scenario.

Every cloud has a silver lining:

In every bad situation there is an element of good

For an article title you can also shorten it to just Silver Lining.

You can describe a person who spreads light, hope or happiness in a negative situation as a ray of sunshine.

There is a phrase gold among the dross (where dross means worthless matter, and especially waste material in smelting metal). There are many variations on this phrase, including gold from dross, gold to dross, and perhaps most frequently, dross into gold.

This is derived from a Biblical passage in Proverbs 25 which actually refers to silver being separated from the dross. There are many literary references to silver/dross metaphors, but they appear to occur much less frequently than gold references, as reflected in this ngram.

If you are looking for a transformative term or phrase, there is the concept of an ugly duckling based on the tale by Andersen. The story describes an ugly duckling who is an outcast, but grows to be a beautiful swan.

A diamond in the rough

Here rough refers to the minerals (generally worthless) surrounding a diamond. This one has connotations of persistently digging through useless or negative things, to find that one diamond. In this case, the diamond is of obvious value, it’s just hidden among a lot of low-value/negative-value things.

It’s very similar to @bib’s suggestion of gold among the dross, though I have not personally heard that one used often. Certainly Google sees much more use of “diamond in the rough.”

Every cloud has its silver lining

Silver lining doesn’t imply the same amount of effort; it’s more just a positive aspect of something that is otherwise negative. It isn’t necessarily hard to find, but it may be hard to notice or appreciate due to the negative thing it came with. I commonly see it used, for example, when disasters brings a community together, helping one another out; that’s a good thing, but it’s not like anyone would recommend going through a disaster for the purposes of building or strengthening communal bonds.

Since someone said "every cloud has a silver lining" I'd go with "A diamond in the rough"

Meaning that among a bunch of rubbish, I was able to find a diamond. It's used in reference to finding a single good thing where you wouldn't expect it. I don't think it applies directly to your situation.

"Looking at the world through Rose-colored glasses" is close. Meaning, that you're seeing things as being better than they really are.

But there is a word for the type of person you describe. An Optimist is "a person disposed to take a favorable view of things." You could title your article "The Hopeless Optimist"

Context: I need to find a name for an article I am writing - which is about how a person turned everything negative into something beautiful.

You want Polyanna / Polyanna Principal

Also consider Voltaire's Candide: Or, Optimism, which satires the idea that this is "The best of all possible worlds" from Leibniz's optimistic conclusion that our Universe is, in a restricted sense, the best possible one that God could have created.

Candide and Professor Pangloss espouse this belief.