There's no particular rule of thumb for sizing stair stringers/ carriages/housings that I'm aware of. However you presumably have a current housed stair to use for comparison purposes - so sizing the narrowest part of a cut-carriage to equal the depth of the housing should work. Otherwise you could use the floor joist tables from the Building Regs to give you a ballpark figure for narrowest point on a cut-carriage.
Are you planning open risers and is that your motivation for changing the design? Be aware Building Regulations has quite a bit to say on stair design.
I'd recommend either one of the general joinery textbooks that cover stairs (e.g. Porter & Rose) or "Basic Stairbuilding" - Scott Schuttner
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