What is the likelihood that someone will be allowed to stay in the US after being picked up by ICE?

By not attending school he is out of status and deportable. He should obtain counsel ASAP who can discuss possible voluntary departure and represent him in court proceedings. If your daughter and her boyfriend are ready to get married that would most likely allow him to stay and she can petition for his permanent residency (green card); but I would never recommend she marry unless she is ready and sure this is what they want to do. Any marriage after deportation proceedings have commenced will be scrutinized carefully to make sure it is being done in good faith. We would be happy to represent him or advise the couple if you wish to consult with me. Contact me as indicated below.

Since he violated her F1 status he cannot change to another non-immigrant status. Also, since he did not attend school for 12 months its unlikely that USCIS would agree to re-instate his F1 status. The only way he could be able to remain in the U.S. is he and your daughter (who I assume is a citizen) get married. If that occurs she can petition him, and if approved he can file for adjustment of status before the immigration judge. However, the marriage must be bona fide and not entered into merely for immigration purposes.