You are creating a SharePoint blog and initially customize it, in order to meet your requirements in terms of design, layout, etc. Please be aware, that you should never remove the Blog Tools Web Part from the pages within your site. Once the Blog Tools Web Part is missing, all links to blog posts and category filters are instantly broken. The pages need it in order to work properly.
How to hide the Web Part properly?
So what should you do, if you need to remove it anyway? Minimize it! Sounds easy. It is easy:
Put your page in edit mode and change settings of the Web Part:
Change the Chrome State:
Set the Chrome State of the Web Part from Normal to Minimized
This approach keeps the Web Part on the page (requirement for the blog to work properly), but yet it is not visible for blog readers any longer.
How to fix your blog, if it is broken
So what should you do, in case you just deleted the Blog Tools Web Part?
SharePoint 2013/2016: Add the Web Part from the Web Part gallery
SharePoint 2010: Restore a recent site backup. The Blog Tool is usually not present in the Web Part gallery.