What are the common causes of on and off itch and redness of umbilical cord of my 2 year old son?

Hard to say: It's possible that there is a little irritation (dry skin, insect bite, moisture from bathing) and then he starts playing with the belly button, so it creates more irritation. You can try some hydrocortisone 1% cream (over-the-counter) and see if it improves. If not, see his doctor.

Sweat, water, bites : Many of these act just like the skin folds of the body, trapping moisture and from bathing or sweat. Soaps can also be trapped. The salt in the sweat causes itching, the itching aggravates the issue by irritating or tearing skin cells, that are even more irritated by sweat. A good cleaning followed by alcohol will dry out the area. Bug bites also trigger this problem.

Skin irritation: And dryness. You can clean it with any ointments applied on a Q-Tip and work your way around the navel ( not umbilical cord ) to loosen up some dirt and moisten the area. If it looks raw or skin is no longer intact, you may use neosporin antibiotic ointment . If infected , you need to go see your Pediatrician.