How to compute Integer Hash of a string
Some useful criteria for evaluating string hashing functions are discussed here:


Performance in Practice of String Hashing Functions
Choosing Best Hashing Strategies and Hash Functions
Evaluating Hash Functions
Hash Function properties


There's an empirical comparison of some popular string hashing functions here which looks at some of the criteria you mentioned. The appropriateness of various hashes would depend on the application and data characteristics. Uniformity and efficiency are usually the major concerns.

If you know the input set in advance, you can generate a perfect hash function that is collision-free. Something like CMPH or gperf can automatically generate a PHF for a set of strings.

If the input isn't known in advance, it still might be biased toward a particular distribution that makes one hash more favorable than other. Benchmarking can be useful.

For a cryptographically secure hash, you'd want something that has good avalanche effect characteristics.

Bernstein's hash is often used in string dictionary data structure implementations, but of course there are many to choose from. (Also MurmurHash.)

I misunderstood the question - this is for integers that are word-sized. For a secure crypto hash like SHA-2, see Kaveh's comments.