You can register the foal half arabian no matter if your mare is not pure bred or registered. The Arabian Horse Association allows registration of Purebred Arabians, Half Arabians (Really any percentage of Arabian blood, as long as one parent is a registered Purebred Arabian.) and Anglo Arabians.
That's one of the reasons I like the AHA so much. It offers an opportunity for foals that would not be registerable if they were another breed, to get papers.
Proud owner of ~Mana: 6yo Arabian gelding~Pearl 13yo Arabian~Danzer 14yo Arabian mare~ Tiny mini filly