We first need to identify the master page that our site is using. We then need to find the HTML element that contains the site brand name. For SharePoint 2013, the brand name is contained in the following HTML element:
On the master page, this is generated from a server side control with this mark up:
<SharePoint:DelegateControl id="ID_SuiteBarBrandingDelegate" ControlId="SuiteBarBrandingDelegate" runat="server" />
document.querySelectorAll('div.ms-core-brandingText').innerHTML = 'All About .NET';
The snippet is selecting all the div elements with the ms-core-brandingText class. There should only be one element with that class name. We then just set the innerHTML to the text that we need. Make the change, save the master page, and refresh the site. It should now look like this:
A drawback about this is that it needs to be done in each master page that your site is using and for each sub-site.
I hope this helps.