Can there be any side effects from a henna tattoo?

Yes, never ever get a "black henna" tattoo. The chemical used to make "black henna" (which isn't actually henna at all) is extremely toxic (I believe it's called PPD).

Also, traditional henna paste can cause skin reactions such as dermatitis, depending on the sensitivity of your skin. It's also possible to have allergic reactions to the oils and other ingredients used in the paste. It's best to test a small amount on your skin before having a large area worked on.