This is what I think:
General, generic memories reside in the mind like they always do. This explains why The Org has memories of their past life without having a heart.
The truly important memories...the memories that allow someone to never ever forget their friends and things important to them, those reside in the heart. It's like remembering the people you love with your heart.
That's what I think, anyway.
Actually, all memories are generally stored in the heart. It's only when a Nobody is created that the memories are released from the heart to the Nobody (in what we can assume to be the mind).
That the Nobodies get memories is fact based on the Secret Ansem Reports and Another Report:
Apart from Naminé, Nobodies retain their memories of their time as humans, but Sora's Nobody, Roxas, has lost Sora's memories.
A Nobody's main characteristic is that without a heart they use memories to form the parts of a personality and emotions. For them, memories have become an essential element.
So how do we know that the Heartless lose the memories and the Nobodies don't simply have a copy of them (I was in an argument with someone over this once...). Simple. Look at Sora. When he turned into a heartless, he described the process of forgetting absolutely everything (all of his memorries were going to Roxas, but when he was regenerated by Kairi's light, Roxas lost those memories and they returned to Sora's heart).
I think it's safe to say that the memories are in the heart unless that heart is lost to the darkness of the heartless.
While it is interesting, the problem I have with this theory is that it's similar to the Heartless. Heartless are comprised of hearts without memories.
By the way, for anyone reading this, I'll be on a family vacation for the next five days just to give a heads up, so I won't be here during that time. (I also have to get off the family computer now since others have been waiting to use it. Unfortunately my laptop was infected with spyware yesterday and can't even reach the desktop :[ ... )