For SHA-256, FIPS 180-4 states the following on the constants:
These words represent the first thirty-two bits of the fractional parts of the cube roots of the first sixty-four prime numbers.
If one were to change some small number of those constants, say $n \leq 8$, to suitably "random" values, would this significantly weaken the cryptographic strength of SHA-256? The intent is simply to use a "non-standard" version of SHA-256.