rbacon

Joechima,
The first step is not a waiver, but is an I-130 immigrant petition submitted by a U.S. citizen wife or child age 21 or older. The petition will progress to a point where additional evidence will be requested, or it may get all the way to the NVC, and then perhaps the Embassy in Nigeria. If the Embassy interview results in a denial, the reviewer may recommend an option to apply for a "Application for Waiver of Grounds of Inadmissibility," I-601, or "Application for Permission to Reapply ...After Deportaiton or Removal," I-212. My experience has been with the I-601 form, once successfully and once with a resulting denial because the applicant had been removed several times.
For a return of criminally-deported persons, a 10-year+ absence seems to be the starting point for establishing eligiblity, but it really depends on the seriousness of the criminal history.
--Ray B

DWheels,
The children must be aged 21 or older (and U.S. citizens) to petition a parent.
--Ray B