Is this a self rolled hashing/obfuscation method? Can you recognize the pattern?
I highly suspect this is a self rolled, or at least very outdated method. It is very weak by modern standards and should not be used to protect anything important.
Short passwords could be cracked with no more than 2^64 brute force attempts, which is quite possible with even a normal computer. If both halves of the result are independent, even with fairly long passwords, 2*2^64 brute force attempts could crack it (so 2^65 attempts). There are likely further weaknesses in the algorithm that make it weaker than described here.
Another interesting point to test would be 2111111111 and see if the second part of the result remains the same. If the second part doesn't change, this is definitely a weak algorithm.
Edit:
Seeing that the results of the 2111111111 test are the same for the second half, this is definitely a weak algorithm and should not be used to protect anything sensitive! I have included the relevant comparison below:
1111111111 : FIcx3a00R4evxqEuQkZZtg==2111111111 : GPwnH80qEACvxqEuQkZZtg==
Edit:
Another interesting test would be what Ricky Demer suggests below:
The next thing to check is that it handles the 8-byte blocks identically. What are the "hashes" of aaaaaaaabbbbbbbbc and bbbbbbbbaaaaaaaac? – Ricky Demer Feb 25 at 6:44
If it handled the 8-byte blocks identically, it is even worse for even long passwords, no matter what the length. It would require just 2^64 brute force attempts as pointed out by Neil in the comments. Once someone made a table of what each possibility calculates to, this algorithm would be practically useless. There are probably even more weaknesses to this algorithm remaining to be discovered.
Bottom line:
I do not recommend using this algorithm regardless of the results of the last test. We have already seen enough to know it is weak. Something with a higher bit encryption / hash scheme, salt, long computation time etc... would be much better, and I would recommend going with one of the existing thoroughly tested methods. Self rolled encryption / hashing has not had the benefit of extensive testing that most of the main stream methods have.
The fact that the length of encoded strings depends on the length of respective passwords suggests a much worse weakness: passwords seem to be obfuscated rather than hashed.
This means than anyone who knows the internals of the system may be able to instantly decrypt any passowrd! The implications are numerous: you have to trust developers who ever worked on the system, the source code has to be kept secret forever (that's why you're reviewing a black box) etc.
It doesn't really matter how good the obfuscation algorithm is (and 264 isn't exactly stellar). It's broken by design.
Kerckhoffs's_principle obviously applies: A cryptosystem should be secure even if everything about the system, except the key, is public knowledge.
This in itself should be enough for you to say "I cannot approve this system without more information."
From your examples, it seems that the input is divided into blocks of 8 characters/bytes, each block is hashed individually and the outputs put together, then the entire result is base64 encoded.
This is not good. Most passwords that are more than 8 character are just a few characters longer. This means an attacker can crack the second block very very easily. And when you know the second block, you have a strong hint as to what the first block can contain.
The crypto community discovered this decades ago and will no longer hash blocks independently. That means these programmers are NOT up to date on crypto.
The only good news is that a single-character change in the password changes its entire output block. This probably means that they are in fact hashing the passwords and not just obfuscating them. Still, with the above weaknesses the worlds best hash function cannot save the algorithm.