to be noncontingent, we can agree, means that the keys to existing are within your nature and not anothers actions. Now, contrary to what Aristotle believed, prime matter or pure potentiality is nonbeing by definition. It is that which holds no actuality, but actuality is any aspect of attained existence. As such, prime matter is nonbeing. Why is that important? Because that fundementally means that actuality is one for one the exact same as existence. So any being which is not pure in actuality and therefore can be reduced to potentiality cannot be eternal through their own nature as that would require ever lasting being which is beginningless, and endless. But their being can have beginning, as it is not pure act and thus able to be caused, and it can have an end, as whatever act it does have can be reduced to potentiality oncemore. Thus, a being must be complete in being to be noncontingent.
Second, there is more to existence then simply the fact that one exists. There is the manner by which one exists. These two things are not disconnected however, for to exist is to exist, manner and all. Any property of the way you exist describes how you are existent at all over being non existent (as both are simple actualities in the end). For example, an immutable being is an eternal and infinite being, for both correspond to each other perfectly, as to be immutable is to hold no change and thus no potentiality, and an eternal and infinite being is simply the complete opposite of potentiality. Eternal being and infinite being must be the same too, as to be eternal is to be unending and unbeginning, but such is in correspondence to an infinite being who s(name removed by moderator)ly is actuality. Now, given that, we know the mind cannot be eternal for it is limited in faculty if it cannot hold memories, as you claim, and even if it did it still lacks in knowledge of all aspects of existence like a pure act being should. Now, to be limited is to be in correspondence with a temporariness over eternality, as to be eternal can only correspond with infinitude. Thus, one who is limited is not eternal without external assistance.