For existing sites your first resource should always be the raw server log data. Forget third-party analytics tools — look at what browsers are actually telling your server about which search engines they came in from and what they were searching for.

You can also try to use server log analysis tools, such as Webalizer but they have limitations just like Google Analytics and its competitors.

For specific search engines, try these tools;
BING
Microsoft’s Keyword Research Tools

Google

• AdWords Keyword Suggestion Tool
• Google Trends
• GoogleInsights for Search

Yahoo!
You have to sign up with their Small Business Service to use their tools.

WordTracker
This is the oldest 3rd party keyword research tool that I know of. WordTracker gets its data from Dogpile and Metacrawler.

Other search services offer less insight but you can at least get an idea of what is most popular with their audiences.

Ask
Ask IQ (Interesting Queries)

Lycos
They seem to have converted their popular Lycos Top 50 report into a Tag cloud on the home page.

There is a horde of SEO tools out there that try to suggest keywords. If you cannot determine where their data comes from they are probably of no use to you.

Compete offers some keyword data based on ISP log data they purchase. I cannot tell you how reliable the data is.

Hope that helps.

As an SEO who is constantly running campaigns with limited budgets, I find it hard to focus efforts on the smaller engines. I find it hard mostly to stay up on what the smaller engines find important for ranking, and how they differ from Google.

Could you point me to any resources, possibly from your old SEO Theory Blog (god bless it) that might get me started on the knowledge game?

Lastly, thank you again for holding the line in an industry were most follow and repeat gimmicky solutions without research.