I'm wanting to move the origin (widget) to the center of this object. The object (mesh and bones) was appended from a previous animation, and it looks the same in my animation. I have tried shift ctrl. alt C > origin to center of mass. Also I noticed that when I shift S cursor to selected, the cursor goes to the widget (origin). I have a feeling this happened because I moved the object with it's root bone rather than the x,y,z arrows. The Root bone is sticking out out the center of the objects back My animation seems to be working fine with the origin being offset as it is. Just out of curiosity, when I render the animation could the origin being offset like this, hurt anything?
I think your problem is this one:
Your armature has an origin, like any object. If you move and animate the whole armature in Object mode the origin will follow, because its position is linked to the object. But if you go within the object in Pose mode and begin to move the bones, it won't affect the origin position in any way, because the origin is linked to the armature position, not to the bones position within the armature, so with some animations the bones can move far away from the origin.
Generally, when you animate in Pose mode, you don't move the root bone, so this bone will stay close to the origin. But if you begin to move it, for a reason or another, it may also move far away from the origin.
If you want to see what is the real position of the bones within your armature, relatively to the origin, go in Pose mode, select all your bones and press altR, altG and altS, or in the Properties panel > Data > Skeleton, press Rest Position, or switch to Edit mode... you'll come back to the rest position and you'll see that the bones will probably come back close to the origin, so you'll probably won't find any reason to move the origin position.
Now if, in rest position, the origin position doesn't please you, which in my opinion is probably not your problem here, you can change it: Switch to Edit mode, then, like for any other object you need to put the 3D cursor where you want the origin to be, then shiftctrlaltC> Origin to 3D Cursor.
If you just want to move the object's origin (pivot) to the geometric mean, this option is available in the Object -> Set Origin menu. This is similar to how it was in 2.79 except now the origin has its own menu, whereas before it was under Object -> Transform.
If you want to move the cursor someplace and then snap the origin to the cursor, the same shortcut for the cursor menu in 2.79 also applies in 2.80. Press SHIFT+S and choose "Cursor to Selected" (where your transform pivot option is already set to "Median"). You can then set an object's origin to the cursor location.