Assuming two things to be true, that we aren't allowed to propose new track infrastructure (no new lines) and that we aren't allowed to propose new services (that each station opened has to fit into an existing service pattern) then the list is actually very easy.
1. Winchburgh.
This station has a formal design already agreed, and the mechanisms for funding it exist, but haven't yet been triggered. Slot it into the Dunblane services for an easy 2tph off the bat, with extra calls in the peak time Edinburgh to Glasgows. It's about as easy an opening as you can get.
2-7, in any order.
Abronhill
Bannockburn/Stirling South.
Bonnybridge
Bucksburn or Hilton (Aberdeen North)
Cove
Dysart
These all have a service that could be easily stopped, giving a minimum of 2tph right off the bat. They all exist on lines with a relative degree of timetable flexibility, meaning that you could slow down those services and add in an extra stop in each direction without scuffing turnaround times or needing an extra unit.
The first three all have had planning processes started and paused, meaning they've at least proven to be technically feasible, although not necessarily economically so.
Cove, Dysart and Bucksburn are somewhat reliant on future electrification allowing journey time improvements, so you don't slow down end to end journey times by too much, but they're all possible without.
They're all in economically underproductive areas, which buys you points in the economic growth category. They're all less than thirty minutes journey time from the nearest major city centre station, meaning that they get you points in the connectivity category. They're all in areas with poor to moderate public transport provisions, again boosting connectivity.
The exact ordering is probably inconsequential, but again I'd think it likely that reopening stations on electrified lines is altogether better, as you can reduce journey times at the same time as increasing capacity, so I reckon that Cove and Bucksburn would be the last to happen of the bunch (additionally Aberdeen council are generally seen as being unsupportive of rail infrastructure)
8-10
After those six it gets a bit more nebulous.
Cumnock isn't a terrible site for a station, but any station would be underserved with 1tp2h in each direction. Cumnock itself as a fair old way from Glasgow, and Carlisle just doesn't offer the same economic pull. You run very quickly into the fact that economically, any station would be a basket case, even if it would have very positive scores for connectivity and deprivation.
Grangemouth would be a great place for a station, but it's a terrible site for a station. The infrastructure isn't there in the way it would need to be, with the services going to Glasgow and not Edinburgh. Any services to Glasgow would be significantly slower than just driving to Polmont. The station would be in a terrible site for the town, and you'd need more of an infrastructure intervention than just a simple platform.
Cambus would tick a few boxes, get an even 2tph, but it's not that far to Stirling, and there's not much there to connect with a station.
Newtonhill, north of Stonehaven, would be helpful, and 1tph in each direction would be a good boost to the town, but it's a small place.
Mossend/Ravenscraig would fit nicely in the Glasgow suburban network, but it's a bit uncertain what you'd be building it to serve. Maybe if and when the Ravenscraig redevelopment comes online you could get a station through developer contribution funding, but again it's a bit of a long way from Glasgow.
Portobello would be a nice to have, but seems entirely unnecessary given the patching constraints on the ECML out to Drem. Brunstane, although underutilised, is only fifteen minutes walk away, and it's not like Edinburgh doesn't have an amazing, if slow, bus network. You'd probably only be stopping the North Berwick services there as well, which is still only 1tph because of those same capacity constraints, and 1tph for what would be a city station wouldn't compete with buses to any degree.
Blindwells with a big car park would prove to be busy, but again those same pathing constraints make it difficult, and it's not that far to walk to Prestonpans.
A WCML parkway station at Moffat would again be a nice to have, but you run into pathing constraints and service provision issues very quickly. 1tp2h to each of Edinburgh and Glasgow, if you stopped all TPE trains there, would struggle to make a business case.
Ultimately, my 8-10 would be Cambus, Newtonhill and Mossend/Ravenscraig, simply because they're the easiest to reopen, most likely to not be an economic basket case, and easiest to serve.
