Can I edit my name on a submitted paper if I changed my name during the review process? I am in PhD and about to change my name. I already have a number of publications.
If your paper has not been slated for publication yet, you can probably change your name on it.  I would contact the editor about making the change immediately.  Depending on the system the journal uses, this may simply require the editor to edit your name in single file; or it may entail quite a bit more work, so you want to given the editor and production staff as much time as possible to make the correction.

You will probably not need to provide proof that you have changed your name.  Name changes are not particularly uncommon; they happen when people get married or divorced, and less frequently (but uncontroversially) for other reasons as well.  If you have decided that it is definitely in your professional interest to change the name under which you publish, that is unquestionably your prerogative.