Esta "Not Authorized to Travel"
 

Recently my boyfriend (Irish national) attempted to travel to the US to attend my mother's wedding and a cruise that he already paid for.
He was traveling with an Esta visa with a one way ticket (he didn't know he needed a round trip ticket) he was detained by airport immigration and stopped from boarding the plane.
He wasn't flagged or banned from America for any amount of time but when he checked on the status of his Esta it said he was now not authorized to travel.
He then went to the US embassy to get a b-2 visa but was denied because he already graduated college and was putting off getting a job until he got back home and didn't have enough ties to Ireland to get approved.
My question is what are his options? Can he re-apply for an Esta visa? Or should he focus on getting a random job so he can get his b-2 visa? Any advice would help!

He needs to contact the US Embassy directly and find out his options. He likely go this ESTA cancelled because he booked a one-way flight and isn't sufficiently tied to his homeland (they assume he will just stay and try to find work off the books). Only the US State department can give him more information on his options, but if he was indeed denied a B-2 Tourist Visa, I don't think he has any additional options at this point.

Why on earth he wouldn't have had a R-T ticket is mind boggling. But that is water under the bridge at this point. (he could always have canceled or changed the return flight)

He is most likely out of luck -- as Doug says he needs to speak w/ the US authorities (not a travel website) and perhaps consult w/ an immigration attorney to see what his chances are.

I agree with talking to an immigration attorney if he really wants to come. It appears he either did not know, or did not understand the terms needed to travel under BOTH the Visa Waiver Program (ESTA is just a pre approval, not a visa) and the B2 visa.
He needs to show proof of funds and ties to home (like a job, family, etc.).

Unfortunately he did everything wrong the first time and that will remain on his record. Th eonly way to get a visa (possibly, don't know how much they will require) is proof that he has substantial funds and that he will be returning to Ireland shortly (not only a return ticket, but a job, a lease on an apartment, family he wants to be near, etc).

Working as a bartender for a month and saving $500 plus the price of a ticket is not going to work since he is already on the suspicious list.

Is the cruise trans-Atlantic to get back home? If he had the proper paperwork for it, this would be proof that he is going home.

The only option I see is for you two to get married :)