The Animagus transformation

Sirius could demonstrate the Animagus transformation for Dumbledore. Remember, unregistered Animagi are apparently quite rare - Sirius couldn't have simply faked it. This, as well as the backstory with Lupin that led to the them becoming Animagi (which Dumbledore knows is true, given the Whomping Willow incident) is a hell of a lie to come up with on the fly if you're a delusional killer.

Harry is still alive

After spending a unknown amount of time with Sirius, Harry still lives. This hardly seems like the action of the mad, devoted Death Eater out for Harry's blood. The only witness that Dumblefore has is Snape, but given his incredible bias Dumbledore seems to ignore his opinions. All he knows is that Lupin and Sirius were in the Shack with the trio. Remember, Dumbledore trusts Lupin, yet Snape was accusing him of helping Sirius. Unless Lupin was also guilty why was he with Sirius?

Occlumency

Small point, but we don't know Sirius could use Occlumency. Snape could, but Snape was heavily involved in that whole area of magic, especially given his Death Eater status. Besides which, he'd have to be damn good at it to get it passed Dumbledore without Dumbledore knowing he was hiding something. And speaking of which...

Dumbledore's judgement

Dumbledore is repeatedly shown to be an excellent judge of character - he is almost always able to quickly understand the type of person he is dealing with (besides Moody, but I think that's a special case). His apparent naivety is shown to be anything but - consider his monitoring of Quirrel, or the scene where he tells Snape "you disgust me"...chilling. As Aberforth says, he's extremely gifted in the art of secrets and lies, as he'd have to be when running the counter-espionage movement against Voldemort.

Sirius' betrayal

An important point is that Sirius never defended himself. He was thrown in Azkaban without a trial while (apparently) going temporarily mad. No-one has ever heard his story! Considering Dumbledore's involvement with the Potters, he was well aware of the bond between James and Sirius. One of the reasons for the furious hate of Black was that his betrayal was so unexpected, so unbelievable. Peter is far more likely to have cracked, and his heroic reputation is the product of his fake death, rather than any sentiment he earned while he was alive.

The Finger

It's a tiny but important detail. Sirius was arrested right after the murders - how would he have known the biggest part of Pettigrew they found was his finger? Fudge himself says it, but he was on the response team at the time. I highly doubt that part of the story was widely known (I don't believe it is ever mentioned by any other characters). But Sirius specifically states Pettigrew cut off his finger before faking his death.

The Animagus transformation

Sirius could demonstrate the Animagus transformation for Dumbledore. Remember, unregistered Animagi are apparently quite rare - Sirius couldn't have simply faked it. This, as well as the backstory with Lupin that led to the them becoming Animagi (which Dumbledore knows is true, given the Whomping Willow incident) is a hell of a lie to come up with on the fly if you're a delusional killer.

Harry is still alive

After spending a unknown amount of time with Sirius, Harry still lives. This hardly seems like the action of the mad, devoted Death Eater out for Harry's blood. The only witness that Dumbledore has is Snape, but given his incredible bias Dumbledore seems to ignore his opinions. All he knows is that Lupin and Sirius were in the Shack with the trio. Remember, Dumbledore trusts Lupin, yet Snape was accusing him of helping Sirius. Unless Lupin was also guilty why was he with Sirius?

Occlumency

Small point, but we don't know Sirius could use Occlumency. Snape could, but Snape was heavily involved in that whole area of magic, especially given his Death Eater status. Besides which, he'd have to be damn good at it to get it past Dumbledore without Dumbledore knowing he was hiding something. And speaking of which...

Dumbledore's judgement

Dumbledore is repeatedly shown to be an excellent judge of character - he is almost always able to quickly understand the type of person he is dealing with (besides Moody, but I think that's a special case). His apparent naivety is shown to be anything but - consider his monitoring of Quirrel, or the scene where he tells Snape "you disgust me"...chilling. As Aberforth says, he's extremely gifted in the art of secrets and lies, as he'd have to be when running the counter-espionage movement against Voldemort.

Sirius' betrayal

An important point is that Sirius never defended himself. He was thrown in Azkaban without a trial while (apparently) going temporarily mad. No-one has ever heard his story! Considering Dumbledore's involvement with the Potters, he was well aware of the bond between James and Sirius. One of the reasons for the furious hatred of Black was that his betrayal was so unexpected, so unbelievable. Peter is far more likely to have cracked, and his heroic reputation is the product of his fake death, rather than any sentiment he earned while he was alive.

The Finger

It's a tiny but important detail. Sirius was arrested right after the murders - how would he have known the biggest part of Pettigrew they found was his finger? Fudge himself says it, but he was on the response team at the time. I highly doubt that part of the story was widely known (I don't believe it is ever mentioned by any other characters). But Sirius specifically states Pettigrew cut off his finger before faking his death.