Is it common to have an itchy bellybutton? If so, what can I do to make it better?

Yes. It is common, especially as the skin over the belly button stretches. You can take Benadryl (diphenhydramine) or use Hydrocortisone cream on it to ease the itch if would like.
No. If you have discharge or redness in your belly button, you could also have an infection. If there is anything unusual, see your doctor before trying medications.