What would be required to evolve an animal (non-human) brain to obtain human-level cognition?
Answering the question what is needed for a human brain to evolve starting from a cephalopod brain is hard. However, I can expand on what is thought to have been the driving forces behind the evolution of the humanoid brain (Hawks, 2013): 

The species of the famous Lucy fossil, Australopithecus afarensis (~3-4 mln yrs ago), had skulls with internal volumes of between 400 and 550 milliliters, which is comparable to chimpanzees and gorillas. Australopithecine brains started to show subtle changes in structure and shape as compared with apes. however. For instance, the neocortex was expanding, which is an indication that the evolutionary pressure on higher functions of the brain was important. 

Homo habilis, the first of our genus Homo, who appeared 1.9 million years ago, saw a modest hop in brain size, including an expansion of a language-connected part of the frontal lobe called Broca's area. The first fossil skulls of Homo erectus, 1.8 million years ago, had brains averaging a bit larger than 600 ml. This shows that language development is a likely factor that has driven evolution of the human brain. 

By 500,000 years ago, the humanoid brain reached a volume of about 1,000 ml. Early Homo sapiens had brains within the range of people today, averaging 1,200 ml or more. It is thought that cultural and linguistic complexity, dietary needs and technological prowess took a significant leap forward at this stage, and that our brains grew to accommodate the changes. The shape changes we see accentuate the regions related to depth of planning, communication, problem solving and other more advanced cognitive functions.

Interestingly, in the past 10,000 years of human existence the brain actually shrank. Limited nutrition in agricultural populations may have been an important driver of this trend. Industrial societies in the past 100 years, however, have seen brain size rebound as nutrition increased and disease declined (Hawks, 2013). 

Indeed, one of the driving components behind the expansion of the human brain is increased availability of fatty fish to sustain the large amounts of fat needed in a brain - survival of the fattest (Cunnane, 2006).  

Although the past does not predict future evolution, a greater integration with technology may lead to further expansion of the human brain (Hawks, 2013).

References
- Cunnane, Med Sci (Paris) (2006); 22(6-7):659-63
- Hawks, Sci Am (2013)