Thinking logically, a 'braid mechanic' should be someone who works on (or, more likely, maintains) a braiding machine:
https://en.wikipedia.org/wiki/Braiding_machine
This modern patent refers to a 'braid mechanic position' which, as far as I can infer from the context, refers to a setting when the device is inoperative for the purposes of maintenance by a 'braid mechanic':
http://www.google.com/patents/CN102430905A?cl=en
This link shows the most common occupations of people with the surname 'Drakeford' in the 1881 census:
http://www.britishsurnames.co.uk/surname/drakeford/1881census
It also indicates that the name was particularly common around Staffordshire and Warwickshire, therefore within or close to 'The Potteries'.
http://www.britishsurnames.co.uk/surname/drakeford/1881census
The top occupations include 'Ribbon weaver', 'Trimming weaver', 'Cotton winder', 'Cotton weaver', 'Silk weaver' and 'Elastic web weaver', all alongside 'Braid mechanic'.
Putting all that lot together, my guess (and it's really nothing more than that) is that you should be looking away from potteries and towards the mills, with your ancestor working to maintain the machinery within them.