One Texas congressional representative is exploring a potential but extreme path towards eliminating Donald Trump's refugee ban executive order — impeachment. According to Representative Joaquin Castro, Trump may have "intentionally exceeded his constitutional authority” by ordering Customs and Border Patrol agents to ignore court orders halting the effects of Trump's new regulation (the Trump administration contends that that order was never issued). While impeachment opens up the possibility that the executive order could be overridden by a new president, predictions for the executive order look like it will stay in place for a long time.
There are a few things that could happen to the executive order should Trump stay on as president (though many are predicting that he won't finish out his term even if he doesn't get impeached). There's the possibility that the order could get overturned in court, but the likelihood of that happening is complicated to assess.
Only two executive orders have ever been overturned by the courts, one issued by President Harry Truman in 1952 and one by President Bill Clinton in 1995. Many legal experts have already presented arguments for the unconstitutionality of the ban, but it's just a gamble as to whether the ongoing legal cases will have any ultimate effect.
Another possibility is that Trump himself chooses to rescind or amend the order, a la the "end" of the birther controversy. It seems really implausible at first that Trump would ever back down like that, but actually kind of makes sense. As long as Trump could figure out some twisted way to come out of it looking like a hero, even though he caused the problem in the first place, he'd probably consider it. The move would appease the left and maybe buy him some political capital with Congressional Democrats chomping at the bit to block his Cabinet nominations, but it would enrage his supporters who have been praising the ban.
In the grand scheme of things, the order will probably stand for the time being, most of the effects will expire at the end of their time limit (90 days for the seven countries and 120 days for refugees), and people will have to lobby Trump to get the permanent freeze on Syrian refugees removed. Protest is the the best recourse for most people, but if you have any legal expertise contact organizations like CAIR and local ACLU chapters to see if you can be of any assistance. Otherwise, keep donating, keep protesting, and keep speaking out.