The short answer is yes .... but... you must:
1. Show physical presence for 10 years;
2. you have qualifying relatives,
3. who would suffer unusual and exceptional hardship to your qualifying relatives and finally,
4. you are a person of good moral character.
The trick is that you cannot complete an application form and submit it to the Immigration Office. This relief is only available if you are put in removal proceedings.
I hope your child feels better.
More information is needed. You will not be eligible for permenent residence based on that status alone. Are you married to a U.S. Citizen? You may be eligible for humanitarian parol if you need to stay with your child while they undergo medical treatment, however this is not any sort of permenent resident status. Has your visitors visa I-94 expired? If you have overstayed more than 6 months then you will be barred from reentry for 3yrs if you overstay for 1yr or more you will be barred from reentry for 10yrs. You should speak with an attroney to help prevent this situation.
Andre Olivie, Esq.