I know that this post is over 6 years old. However, since it is in the FAQ section, I felt the need to comment.
I am very glad the child in question turned out okay!
But... in case anyone comes across this thread with a similar, more urgent, problem... putting water or saline on a fragrance oil or essential oil "burn", whether on skin or a mucous membrane such as the eye, is one of the worst things you can do.
Water and oil repel each other, and the water will not attach to the oil. This means that it will not effectively flush the damaging oil; all it will do is spread the oil and drive it in further, making the problem worse.
The first thing to do is flush the area with a benign carrier oil, such as jojoba or almond oil. Olive oil, a more common household staple, can be used. Whole milk, with its high fat content, can even be used in a pinch. Doing this will actually dilute and flush the harmful oils.
Once the fragrance oil has been flushed completely, feel free to use water or whatever you choose, but please don't use it initially.
Hopefully this will help someone!
