Most part of my result is as same as orginal version. Can I make a paper? I am a PhD student in mathematics. Recently, I could extend some old result to higher dimensional. I want to make a paper. But I doubt that is a good result or not, because most part of my proof is the same as the proof in the orginal version. In fact, what I did: I try to understand which part of the proof does not work for higher dimensional. So, I replace them by  higher dimentioanl theorems. Do I neglect my result and do not make a paper? Or I have to do it?

I checked that no one did it before me.
Normally extensions to existing work are valuable. I suggest that you take advice from your advisor if there is any to give. 

But you can also write it up and submit it to a journal. Let the editors and reviewers decide whether it is worth publication. It would be a mistake to just assume that it isn't without feedback from other mathematicians - such as reviewers. 

This is true since, as you say, the original proof doesn't work for higher dimensions and the original authors may not have contemplated the extension.