I've looked at several hash function specifications, and they all emphasize the need for big-endian byte ordering. Is there a reason for this that has to do with security, or is it simply convention?
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Probably you just looked at the SHA-family of cryptographic hash functions as others (like MD5 and the RIPEMD-family use little endian (see for example this table for the compression functions).
The emphasis on a certain byte ordering is purely functional and not relevant for security (see my 2nd comment to your question).
If we look at the Advanced Hash Standard finalists, we find that 4 of the 5 finalists do not insist on a big-endian byte ordering:
Here, we don't see any strong bias. I suspect jug is correct; you just looked at the SHA family of hash functions.