Aragorn in Moria?

[The Might]

Why...is a really good question...
My opinion is it had something to do with finding out information on Sauron's plans, just like his journeys to the South and East and the outskirts of Mordor.
I'd like to use this quote, already posted once in this thread by Morwen:

He rode in the host of the Rohirrim and fought for the Lord of Gondor by land and by sea; and then in the hour of victory he passed out of the knowledge of Men of the West, and went alone far into the East and deep into the South, exploring the hearts of Men, both evil and good, and uncovering the plotsand devices of the servants of Sauron.

LotR - Appendix A (v)

We know that already in 2480 Sauron began to people Moria with his creatures, also as mentioned in the Tale of the Years, so it makes sense that Aragorn went in to take a look and see what the situation was.