Different themes and different versions of concrete5 handle the Site Name...well differently.
If you're using the latest version of concrete5 and one of the default themes (Plain Yogurt, Greek Yogurt, Green Salad), you can put anything you want in that logo/site name space using Global Stacks.
Go to Dashboard > Stacks & Blocks > Stacks. This will open up a Stacks overlay. Under Global Areas, select “Site Name” and then click the block you want to alter (with a new site that has been filled with sample content this would only have one block of Content “My Site”). Select edit from the popover menu, change your site name and click save. Now when you return to the website you will see your new and improved Site Name.
You can also get into the files that power your template on the backend and change it there.
If you're using concrete5 version 5.4.x and one of the default themes (Plain Yogurt, Green Salad), you can put anything you want in that logo space using a scrapbook:
Go to Dashboard > Scrapbooks > Global Scrapbook.
Edit the the My_Site_Name block.
You can insert an image here, delete the content entirely, whatever.
If you are using an older version of concrete5, or you are using some of the themes from our marketplace that were made before this change, you'll have to do two things:
1) go to Dashboard > Settings and change your sitename.
2) go to Dashboard > Settings > Debug and clear your site cache.