How to remove a file from a "Choose file..." dialog?
I guess you are refering Drupal 7, as Drupal 8 has by default that option.



D7

What Drupal (D7) Provides,

Step-1: File/Image uploading option in node.


  


Step-2: Example Case: Suppose I uploaded wrong image instead of IMG_0065.jpg, I want to upload IMG_0068.jpg.


  


Step-3: Just Re-upload the file on the previous file, the new file will be loaded and the previous file will be removed.


  


Step-4: Image file uploaded and now als0 you can remove as Remove Button is there. 


  


So, In short you want REMOVE button at Step-2, But the default file uploaded do not have that option. That is the fact. But if you manage, Drupal provide the option of replacing, which is quite similar option of removing.

Solution using other file uploader Module

Now, instead of using Drupal's Default File uploaded feature, you can download new module called, Drag &amp; Drop Upload.  Installation: For this also like default file uploaded, you have to create add a field in content type and in widget option you need to choose Drag and drop instead of file.

Now, following is the image of drag and drop in action, now you can see after just browsing, you can see X mark besides file. Which is the the Step-2 option which was missing in above default setup. 


  


Further Reading: On this link there is a detailed comparison of all type of File Uploading Modules, you can refer more from there: 
Multiple file upload for node




D8

By default Drupal 8 has this option as shown in following image.