What you are looking for is an editable ContentControl. If you have this - you can put it wherever you need to edit anything, including the TabItem Header.
Your question got me thinking whether this will actually work and I decided to test it. I created a sample project with an editable ContentControl. It is just like a regular control but on double-click the content presenter is replaced with a TextBox. It still needs a few things but it can be a starting point for you:
I the example I use DataBinding, though you can create the TabItems yourself and set the HeaderTemplate to the sample DataTemplate in the project.
Improvements may include: Committing change on enter key, using the TextSearch.Text attached property for visual objects, etc.
I hope that this will get you going, also if you come up with something interesting, please share it with us.
All the best,
the Telerik team
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