North Carolina:
NC-01: Rocky Mount and Greenville
NC-02: Raleigh
NC-03: Outer Banks
NC-04: Durham and Chapel Hill
NC-05: High Country and Gaston (and Foothills could work too, but it doesn't include Catawba the largest foothill county so idgaf)
NC-06: Greensboro and Winston-Salem
NC-07: Cape Fear and Johnston
(Wilmington and Johnston or Southeast and Johnston could work too, but this, the 8th, 9th, and 10th districts are such abominations, it's hard to come up with 'good' names so I'll just go with names that cover the geographic extent of the districts)
NC-08: Concord and Fayetteville
NC-09: South Mecklenburg, Union, and Sandhills
NC-10: Lake Norman and Outer Forsyth
NC-11: Smoky Mountains
NC-12: Charlotte
NC-13: Salisbury, Lexington, and Burlington
These are great district names for what surely must be difficult map, but I will nitpick and say:
-Outer Banks isn't a great name for NC-03; the actual Outer Banks are pretty sparsely populated. The real population centers are in cities on the mainland, like Jacksonville, Kinston, New Bern, etc. Maybe Eastern North Carolina or Albemarle-Jacksonville? Don't know this area super well.
-You could probably just call NC-09 South Charlotte and Sandhills.