The O-1 evidentiary standard lives in 8 C.F.R. § 214.2(o)(3): extraordinary ability in the sciences, education, business, or athletics is shown by a major internationally recognized award or by at least three of the enumerated criteria; extraordinary achievement in film or television has its own showing. This is a different test from the EB-1A green-card standard — there is no Kazarian-style "final merits" second step written into the O-1 regulation — so an O-1 denial is challenged against the O-1 rule on its own terms, not by importing immigrant-visa case law.
Two O-1-specific requirements are common denial points. The consultation requirement (8 C.F.R. § 214.2(o)(5)) generally calls for a written advisory opinion from a peer group, labor organization, or management organization; a denial premised on a missing or unfavorable consultation is often answerable on a motion or a refile. And the petition must be filed by a U.S. employer or agent with a described itinerary of events or activities — denials on the employer-relationship or itinerary rules turn on documentation that can frequently be supplemented.
The administrative remedies are the same ones that follow any I-129 denial. A motion to reopen (new facts) or a motion to reconsider (legal or policy error on the existing record) goes back to the service center on Form I-290B under 8 C.F.R. § 103.5, within 30 days of the decision (33 if mailed). An appeal runs to the Administrative Appeals Office under 8 C.F.R. § 103.3, also on Form I-290B, for de novo review. Only the petitioner — the employer or agent — may file; the beneficiary is not an "affected party" with standing.
Federal court is available too. A denial that misapplies the O-1 regulation, ignores probative evidence, or fails to give a reasoned explanation can be challenged under the Administrative Procedure Act (5 U.S.C. § 706(2)(A)) as arbitrary and capricious, and for most petitions an AAO appeal is not a prerequisite to suing. Litigation is the right tool when the defect is legal error on the record the agency already had; where the real problem is a thin record, a premium-processed refile is usually faster and cheaper.