Abbeyhill has a couple of problems.
It was built as an intermediate stop on the way to Leith Central and Trinity / Granton, but it's a dead end now.
You could leave it as a dead end, but that would preclude access for trains running from the east (North Berwick / Dunbar / Brunstane and in future the Borders). It is in no way practical to bring trains to a halt on the ECML approach to Waverley to then reverse them back into Abbeyhill. That is just never going to happen and I think rail regulations possibly even preclude such a dangerous manoeuvre. So you could extend trains running from the west through Waverley and terminate at Abbeyhill? But there is limited through capacity at Waverley, as the bulk of platforms are blocked by the central ticket concourse. It isn't viable therefore to run many (if any) trains originating in the west to it.
So the "obvious" solution is not to use it as a dead-end and would be to reconnect the loop under the road, and back onto the East Coast Mainline. The remains of this line are singled and signalled as a single block for the "binliner" trains. You would therefore need to pay for the line to be doubled, to be resignalled for diverting trains from east off of ECML through Abbeyhill before they reach Waverley and you would also need to electrify it if you wanted North Berwick trains to be able to access.
Alas in practice to my mind it seems quite a bit more complex and a great deal more expensive than simply popping a couple of platforms and some access in at Abbeyhill.
In my personal point of view, an intermediate station directly on the ECML might be more useful, there is plenty of space at Craigentinny or site of old Portobello station.