As with any serious restoration considerations, it will in Portugal, like in any other country, be a matter of well organized publicity, hard work from the Royals, tieing bonds with important political aspects of society and generally raising awareness of their own person, the nations history and the advantages of monarchy.
15% support for the monarchy is quite strong in a 100-year old republic, and the Duke of Braganza is quite a familiar figure in Portugal, respected by politicians. If he were to seriously seek a restoration for his family, it would mean getting himself out there, alongside his family and heirs in particular, and see the numbers rise with hard work done.
The Duke said himself in an interview in 2007, that 'a King is always a better head of state than a president'.
Obviously, a monarchist would agree from the get-go, but the job is to instill that sentence in the heads of the average Portuguese person.
Incidentally, I was in Portugal 2 years ago, and stayed with an extended family of friends for a few weeks. One night across the dinnertable I raised the issue of the past and future of Portugal, and after a lively debate, not a single one of them wanted to keep the republic.
The old monarchy is gone, but the Royal Family isn't.
In modern days, the advantages of constitutional monarchies, especially in Europe, is not a hard thing to sell. It's actually quite easy.