Usually our thoughts have a certain content or object, but in some way one can say that it is the person who has the thoughts. But is there also a mental thought involved of this thinker as a thought? How is this named in Buddhism? Is that 'ego', also as a thought possible? I hope it is clear what I mean....
Perhaps you're thinking of various words which are translated as "mind" or "consciousness".
I think it's said that the mind is aware of thoughts -- or if not "mind" then "consciousness of mental activity" ... in the same way that "eye consciousness" is aware of sights, and "ear consciousness" is of sounds.
Western philosophy has theories like "I think, therefore I am"; but I think that Buddhist doctrine includes statements such as, "I am not these thoughts, these thoughts are not mine, and are not me."
Because you're constantly aware of them, though (in the same way that a dog who is tied to a post will be constantly aware of the post), you might be inclined to view them as "me".
Or the word you're asking for may be "self-view" or "identity-view", which is a view (or thought) that the "self" exists. Or if it's not a fixed/permanent view, it might be called "pride" or "conceit". See ...
... for some further answers about these topics.