As others have said it won't have a great business case because it is too far from Edinburgh to attract many daily commuters.
Hawick does however have a population of 15,000 or so which is far in excess of anywhere else in the Borders except Gala and it has suffered from the decline of the Textile industry so has a decent regeneration case.
If rail can make a decent fist of serving the BGH (local hospital) and the Council HQ in Newtown St Boswells then it may perform a useful Intra Borders role as well as Edinburgh commuting thus helping the Hawick business case.
I suspect it still won't be a fantastic BCR but with Wider Economic Benefits (WEB) it may just be viable to Hawick.
Hawick could attract commuters with sufficient journey time improvements on the existing line. 55 minutes from Edinburgh to Tweedbank is very slow and could be significantly improved. I think sections of the line are as low as 30-45mph and electrification would reduce the time penalty of having several intermediate stops. The line was built with sufficient clearances on new infrastructure for wires so it would be more straightforward than on other lines. If Hawick to Edinburgh in 55 minutes could be achieved it would be attractive to commuters. Whether any extension is worth the investment is a different matter! An extra couple of miles of double track would be very useful and electrification would be beneficial and might make financial sense though.