I answered these questions based on:
1. what would be appropriate/beneficial for the tasks I think would be most closely related to the job I'm applying for.
2. prior informal research within the actual department I plan to work at, asking the other employees what they need or have been lacking in terms of human resource.
3. the interviewer's body language; encouraging what they reacted positively to, and keeping information to a minimum when I sensed potential controversy.

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I got the job (applied for a marketing/sales position in a computer hardware store, selling mostly Apple products to college kids). My personal answers to the following:
1. Strengths
(I aimed to tell what good values I have indirectly)
+ personable, excel in interpersonal relationships (good communication skills)
-----------------------> I was then asked how am I personable; I mentioned in my last customer-service-related job as a door-to-door sales distributor, I had to meet with an average of 300 customers on a daily basis; and so I can deal with high-pressure situations as such.
+ very detail-oriented (observant/analytical)
+ don't like/try not to give false promises (reliable/trustworthy/honest)
+ see assigned tasks/promises to the end (dependable, dedicated; ref. above)
+ perfectionist-ic; make sure things are done right (good work ethics)

2. Weakness(es. My interviewer asked only for one)
(I stated it matter-of-fact-ly, just to get it out there and make it seem as if I'm not that affected by it enough to hurt my self-esteem)
- perfectionist-ic; tend to over-analyze sometimes just to get things done right
(then I mentioned I'm working on taking it easy, for the most important thing is the company's and customer's satisfaction, and 'it' is getting better now)


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I hope this helps! It's pretty tricky, especially when it comes to describing your pros and cons realistically and constructively. So long you show motivation, dedication, and effort, I doubt you'll do poorly with these questions.