The big factor is MOI (Moment Of Inertia) or the bigger the sweet spot by design the less effect of an off hit putt. More modern putter models have higher MOI creating bigger sweet spots and solidness of contact. The putter design has a equal effect on both directional and distance control. If the player would strike the putter perfectly every time in the center of the putter there would be no need for so many club head designs. Every manufacture has touted their design to influence the roll of the putt in a positive manner. Between the driver and the putter they are the most researched and have resulted in many innovative and exotic designs.
For the player who has trouble hitting the sweet spot, a bigger head with a higher MOI would be quite beneficial. Higher MOI heads are larger with more weight distribution on the heel and toe.
Putter shafts come in straight, single bend or double bend and in materials such as steel, aluminum, and graphite. They come in various flexes and can be center mounted, end mounted and shafted with or without a hosel. Much of this is personal preferences regarding the feel and looks and by far the most important is the feel so the player is comfortable with the choices.
Putter grips are small, medium, large and extra large and also come in various textures and lengths. The bigger the grip the less tendency to use the hands in the stroke but the result sometimes is also a loss in feel. Again it is a personal preference. Those that like to use a longer putter to change the weight or counterbalance, need a longer grip as the need to choke down to their proper length. Below are some commonly used putter grips.
Since I believe that putting is a huge part mental, if a person would go out and design their own putter out of whatever material they wanted so it would feel just right for them the result would be, they would putt better because they are totally committed to the putter. The confidence and enthusiasm in the creation would result in concentrating harder on every putt. I experienced this when I was a youngster and watched my father design an aluminum putter head, had it poured at a foundry, sanded and grinded it and weighted it toe to heel with lead. As a result he was a great with the putter, always made it when it counted. In my collection I still have several he made.
In summary design is a personal preference and I have not met a golfer yet who does not have several putters in backup. I have often heard the quote "If it isn't working, bench it and get the other one".
Getting properly fit removes the question is it me or the putter and then you too can have confidence and enthusiasm in knowing you have a properly fit putter.