JK Rowling at the Edinburgh Book Festival August 15, 2004:
Q:All the paintings we have seen at Hogwarts are of dead people. They seem to be living through their portraits. How is this so? If there was a painting of Harry’s parents, would he be able to obtain advice from them?
JKR: That is a very good question. They are all of dead people; they are not as fully realised as ghosts, as you have probably noticed. The place where you see them really talk is in Dumbledore’s office, primarily; the idea is that the previous headmasters and headmistresses leave behind a faint imprint of themselves. They leave their aura, almost, in the office and they can give some counsel to the present occupant, but it is not like being a ghost. They repeat catchphrases, almost. The portrait of Sirius’ mother is not a very 3D personality; she is not very fully realised. She repeats catchphrases that she had when she was alive. If Harry had a portrait of his parents it would not help him a great deal. If he could meet them as ghosts, that would be a much more meaningful interaction, but as Nick explained at the end of Phoenix—I am straying into dangerous territory, but I think you probably know what he explained—there are some people who would not come back as ghosts because they are unafraid, or less afraid, of death.
And from Tales of Beedle the Bard:
Wizarding photographs and portraits move and (in the case of the latter) talk just like their subjects. Other rare objects, such as the Mirror of Erised, may also reveal more than a static image of a lost loved one. Ghosts are transparent, moving, talking and thinking versions of wizards and witches who wished, for whatever reason, to remain on earth. JKR
Tales of Beedle the Bard - Page 134 - Collector's Edition
This implies to me that portraits are not "thinkers," while ghosts retain their ability to think as they did when they were alive.
That said, Phineas Nigellus seems to have quite a lot of depth to him for a portrait, and is able to plan, spy on the trio (Forest of Dean in Deathly Hallows) and report information back to Snape, act autonomously (going to his portrait at Grimmauld Place to verify Sirius's death in Order of the Phoenix), and like or dislike people on a case-by-case basis.
If I recall correctly, Dumbledore mostly slept while in his portrait and held a few cursory conversations with Snape and Harry respectively. He seemed to lack his in-life brilliance and acumen. Anyhow, JKR answers the question of how intelligent and capable of original thought individuals in the portraits possess.