Sheer brain size is not very important, encephalization quotient is considered more representative and elephant (1.2..2.0) one is only marginally superior to cat (1.0..1.8) and much lower than Homo Sapiens (7.2..8.0).
This method is not universally accepted, mainly because it gives only a rough estimate. Main idea behind this is a certain (large) amount of the brain is used to control and "model" the body, i.e.: it's used for "general maintenance". Since nervous tissue is very expensive, in terms of nutrition requirements, there's a strong evolutionary push to keep it to the "minimum required". A much better approach would be to really try to understand exact functions of the various brain structures. This has been attempted, but attempts have been hampered because, while certain functions are quite clearly localized, other (especially "higher" functions like language) are only pertly localized (even lateralization is far from being clear-cut), many functions are carried out involving large sections of the brain, sometimes virtually all of it. In practice encephalization quotient, corrected with any known "special requirement" (e.g.: dolphins "excess cerebral mass" is largely used to implement their sonar sense) gives a reasonable hint without the obvious anomalies the sheer mass has (sperm whale should be much smarter than we are and women should be less intelligent than males (yes, I know a lot of people is convinced of the latter, but that doesn't make it true!)).
The key to reach self-awareness is language. to date we are the only species who developed one (not a set of signals, but a real language capable of grammatical and syntactic transformations).
I need to expand this to explain I consider a language a construct following grammatical and syntactic rules capable of recursive definitions, so that is able to describe and manipulate concepts not immediately related to physical objects. Regardless of "common usage" I do not consider the simple passage of information as sign of existence of a real language, otherwise bees would have a language (fairly structured too, but inflexible conveying information on a single topic) as most, if not all, animals.
Language creation seems to be a necessity of the human species, when in sizable communities (see: Nicaraguan Sign Language), while humans grown without contacts with other people won't develop one and can't really be taught a language anymore (there have been a few cases of children raised by animals in the wild). This characteristic seems to be what put us apart from other animals and strictly connected with what we call "self consciousness".
This process could be duplicated, of course, if we knew what really makes the difference, but I strongly suspect we still lack the necessary insight to understand what needs to be done.