What is the fastest way to count the unique elements in a list of billion elements?

I would skip the data structures exercise and just use an SQL database. Why write another custom data structure that you have to analyze and debug, just use a database. They are really good at answering queries like this.

I'd consider a Trie or a Directed acyclic word graph which should be more space-efficient than a hash table. Testing for membership of a string would be O(len) where len is the length of the input string, which is probably the same as a string hashing function.

This can be solved in worst-case O(n) time using radix sort with counting sort as a stable sort for each character position. This is theoretically better than using a hash table (O(n) expected but not guaranteed) or mergesort (O(n log n)). Using a trie would also result in a worst-case O(n)-time solution (constant-time lookup over n keys, since all strings have a bounded length that's a small constant), so this is comparable. I'm not sure how they compare in practice. Radix sort is also fairly easy to implement and there are plenty of existing implementations.

If all strings are d characters or shorter, and the number of distinct characters is k, then radix sort takes O(d (n + k)) time to sort n keys. After sorting, you can traverse the sorted list in O(n) time and increment a counter every time you get to a new string. This would be the number of distinct strings. Since d is ~15 and k is relatively small compared to n (a billion), the running time is not too bad.

This uses O(dn) space though (to hold each string), so it's less space-efficient than tries.