Meet with a qualified immigration lawyer
If you did not have counsel at your removal hearing, then you should highly consider retaining experienced counsel to prepare the stay of removal request.
Prepare form I-246
Preferably your attorney will prepare this form for you. However, if you can't afford an attorney, then you need to track down form I-246 from the ICE.gov website.
Supplement form I-246 with ample evidence of hardship
In order to have a successful case, the deported alien needs to provide a lot of evidence to show ICE why the person cannot return to their home country. Generally, extreme medical hardships will suffice for ICE to grant the stay request. The alien should provide letters from doctors, prescriptions, and a detailed affidavit. In other cases, if the alien can show that they are minors, enrolled in school, and that their parents have status in the U.S., then ICE may exercise some discretion and grant the stay for the period of school enrollment.
Attach correct fee of $155
The alien must include the correct filing fee in the form of a money order or cashier's check payable to the U.S. Department of Homeland Security.
Must deliver stay of removal request in person
This is why you may want to have an attorney prepare the case for the alien. The attorney will hand deliver the completed packet to the local ICE office.
Wait for decision
If ICE approves the stay request, then the agency will stamp the I-246 and send proof of this to the attorney or the alien's address. ICE will then place the alien under an Order of Supervision. The alien will then have to comply with strict reporting requirements. However, this is better than being physically deported.