One way to evolve it could be to require it be submitted to the President, for instance, have the ABA and some other civil justice organizations put together a list of convictions that have a high probability of being in error, or where its believed the sentence was disproportional for someone who was actually guilty.
ABA for purposes of correcting issues in their profession, and groups like the Innocence Project as a way of augmenting their work. President should then more or less rubber stamp it, with no ability to add to the list. This would also require some edit to the pardon power so that it could be used in cases where the person is believed innocent – currently a pardon is a pardoning of a crime, for the guilty.
I’m not saying this is a best solution, but just one of likely many ways this function could be preserved. I think it does serve as a valuable check on the powers of the judiciary. With all the nutters and extremists that the GOP have been packing the courts with lately, I expect we’ll need the ability to fix a number of convictions in the years to come.