As Lou Davis says, there is no function of the navel for autonomic function past birth. But it can be argued that there is a musculoskeletal / myofascial function to the navel as it relates to deeper muscular and connective tissue structures in relation to anatomical development and subsequent physiological functional capacity.
The navel is a connective tissue (fascia) locus that is centered around our physical center of gravity, which is why physiotherapists make such a fuss about “core” strengthening. The abdominal muscles and associated connective tissue (including in the area of the navel) constitute a center of support for the viscera and the musculoskeletal (including Lumbar) structures surrounding and supporting it.
This becomes important, in physiotherapy, when the “core abdominal support” has been anatomically or physiologically / functionally compromised. The navel itself doesn’t have muscular components, such as contractive sarcomeres, but it affects the sarcomeres of the surrounding myofascia, including rectus abdominus and external obliques,etc. The navel itself is not the issue, but its location at our center of gravity and the functionality that extends from that center of gravity is the issue.
 Fascia: The Tensional Network Of the Human Body; 2012, Elsevier; Robert Schleip, Thomas Findley, Leon Chaitow, Peter Huijing; pgs. 50-52