For most decks, you really get to choose between 4x4 and 6x6 posts. Some really big decks can go big on posts, not us. In my area, they define any deck over 8' as requiring 6x6.
I think from a looks perspective, as well as strength, most decks should go 6x6 over 4x4. From a lumber cost perspective, the impact isn't massive. Online HomeDepot.com has 8' 6x6 for about $21, and 4x4 for $8. There are downstream costs though.
For my 12 post deck, I only need 6 of those, maybe a couple more for angle bracing. Lets say 8 boards.
Here is my cost impact that I see:
- lumber (21-8) * 8 = $104
- post bases (35-27) * 12 = $96
One possible advantage in 6x6 I see though, is the ability to notch the posts. Obviously you need a little more post in terms of length, but only 1' per post max. Then the trade off is post cap vs. bolt prices, which should come solidly on the side of notched posts. If you have to use post caps though, it adds even more cost!