In you run an analysis on your logs, you could look at how often your brand name came up in the search terms users typed in to find you. Presumably, if they are typing in the brand name "Wigitron", it would indicate a greater brand awareness than if they just came accross you site while doing a search for more general widgets. If the proportion of "Wigitron" searches grows over time, you might consider that reflective of an increase in brand awareness.
you might also consider creating a list of your core brand keyword permutations (e.g. brand, brand.com, brand good, brand bad, etc...) and using a keyword tracking tool to check volume of searches each month. this will give you some indication of "buzz factor"
you might also consider periodically monitoring what the SERP's are saying about your brand. its not just about ranking in top 2 positions for your brand keywords, but what do results 3-20 say about you? are the listings accurate, relevant, positive?
you could run a exit survey on your site that is triggered by users who came through a SERP. if you can get the right adserving logic built in, you might even be able to qualify only users whose search query contained your brand.
Ask them questions geared to understand how they found you, what they were looking for, what is their opinion before/after the visit, etc...