My existing door jamb is 5 9/16 i m ordering pre hung interior doors my cut choices are 5 1/2 ( 5 8/16 ) or 5 5/8 ( 5 10/16 )
What would you do and why or why not on either
If it is a split jamb it will not matter it will gauge itself to the wall thickness. If it is not, and you have an older home with 3 5/8" studs circa 1950-60's with 1/2 sheetrock, you will need the wider jamb. If it is a newer home, you may be able to go with the narrower jamb.
The biggest downside to a smaller jamb, is there may be a gap that can occur at the point where the trim meets the jamb. It can be caulked is it happens. This is still unlikely though since the trim has a relief to allow it to bridge over some unevenness in the assembly.
The biggest downside of the wider jamb is the reverse, a gap at the wall to the trim, but it will only happen at the corners, unless you force it back, which will then open the joint.
I would go with the wider jamb without seeing the conditions, if the walls are out of plumb a little you can use the extra width to tweak the jamb into plumb without going too far behind the face of the wall on either of the four corners. If the wall is too far out, use a plane to get the jamb edge closer to the wall face can be done too.