I don't know what the Name Check exactly is.
Maybe they check government records to see if there
is any activity on your name ever. But the fingerprints are used to check criminal record. Fingerprint check is very quick now, one or two weeks.
Name check requirement is sent to FBI by USCIS just after you submit your I-485. It'll take several months or several years. Many people are waiting for their
name check in this forum.