Best practice, as always, is to not "keep" what is mapped automatically by TestComplete when doing recording or mapping of objects but to spend the time modifying the mapped objects to include the proper unique identifiers and properties. From the screenshot, I see ObjectType is TextBox. So, that's one property I'd keep when editing the NameMapping for the object. I'd then include the idStr as that seems to be the best bet at an additional property for a unique identifier. The combination of those two and the relational hierarchy should be sufficient to identify your object.
Beyond that... this is elbow grease stuff... If you haven't been monitoring and modifying your NameMapping as you go, then you will have to go back and update items. Now, looking at the Alias in that screenshot, it seems obvious that not ALL the objects in the hierarchy are mapped, probably because Extended Find is probably on by default.
So... it comes down to you needing to do work to make sure your NameMapping is sufficiently robust. That seems to be the answer to your question.... that is, unless you have a specific problem you're trying to solve.