You know about Engrish? Well, there is also Ratin! And it's all over the place!
For instance, there is the name of this freethinker's blog:
So, apparently, the person is trying to say "Think for yourself." But cogita tute means something like, "think! you yourself" where the cogita is a direct command, from cogitare, "to think," and the tute is an emphatic form of tu, the pronoun "you" in the nominative singular.
Cogita Tibi would be much better, where the tibi is in the dative: think for yourself, as in think for your benefit.