No, 1 is better. We choose between "a" and "the" based partly on context. Consider this:
The house has a cat in it. The cat is black.
There is only one cat. And in the second sentence, we use "the" because we're talking about a specific cat. But in the first sentence, we're still establishing that context. We really mean "the house has one single instance of the category 'cat' in it," so we say "a cat" for that one.
Similarly, it makes sense to say "the word cat has a t in it." Having established that context, we might use "the" if we continue to discuss that letter that we just identified: "The t is the final letter."
EDIT: As I think about this more, I was puzzled why you might think that having "only one" of something would call for "the." After all, there are plenty of times that there would be just one of something, but "the" would be wrong. "There is a tree in our yard." Just one tree, but we wouldn't say "There is the tree in our yard." (Well, unless we were pointing at it.) I think maybe the source of your confusion is about proper nouns vs common nouns. There is only one Eiffel Tower. Ignoring replicas and models, there is only one "true" one in the world, and whenever we talk about it, we're talking about that one. In Paris, you would not say "Look, an Eiffel Tower!" With proper nouns we use "the" because we've already specified the single thing we're talking about. But cats, trees, and the letter "t" are common nouns. There are many cats, trees, and letter "t"s. We would use "a" when first identifying an instance, of them, and then could use "the" after having focused on that specific instance.