Why did the Joker choose Harvey Dent?

The Joker chose all three (Harvey, the Batman, and Jim Gordon), each in his own way, but only Harvey fell completely.

Harvey was attacked through the loved one, done by kidnapping and murder. The Joker gave the Batman switched locations, because he knew that the Batman would go after Rachel (The way you threw yourself after her), so with one move he achieved a strong blow (soon to be proven fatal) to Harvey, and heavy guilt on the Batman who made the choice who to save, and never forgave himself (the guilt follows him to "The Dark Knight Rises", almost a decade later).

This was done while the Joker was in custody, and by using Jim Gordon's corrupted police officers, making Jim Gordon directly responsible for the outcome.

In effect, Harvey felt betrayed by his colleagues and he has turned on them, but in reality, all of them were victims of the Joker's plan.

The Barman: What happened to Rachel wasn't chance. We decided to act. We three.

Harvey: Then why was it me who was the only one who lost everything?

The Batman: It wasn't.

Bruce also lost Rachel who he believed was going to marry him.

Jim Gordon almost lost his son, although not through the Joker's direct actions. However, in "The Dark Knight Rises" we see the real aftermath on him: the events and his consequential crime fighting obsession cost him his family.

Congressman: Anyone shown him the crime stats?

Foley: He goes by his gut and it continues to bother him, no matter what the numbers.

Congressman: Must be popular with his wife.

Foley: Not really, she took the kids and left for Cleveland.

In my opinion, Harvey was the weak link. Bruce and Gordon endured (although, Bruce almost gave in to the Joker's demands, and later became a recluse). They were mostly cool-headed, fighting crime, while Harvey was more into public theatrics (for example, the courtroom scene), and rash decisions (for example, abduction of Thomas Schiff), while completely unfit to deal with the consequences of the actions that were his, and not just his colleagues'.

So, Harvey was the only one who fell, but all three were attacked by the Joker.