Hash based PRNGs can be surprisingly tricky to reason about. For example, the other common way to turn hash functions into PRNGs (state = hash{state}) often leads to severe flaws from state space collapse, among other issues.
Re-reading your comment above, I had missed that you were assigning the output of the hash function back into the state. I.e., something like:
state = initial_seed
def random():
state = hash(state)
return state
seq_num = 0
def random():
result = hash([initial_seed, seq_num])
seq_num += 1
return result
For what it's worth, I did a (very informal) comparison of MurmurHash3 vs SipHash when I started my approach and found that MurmurHash3 (despite being advertised as passing the avalanche test) gave very statistically biased results. Something that should have generated a uniform distribution definitely did not. Whereas when I tested SipHash the output looked (at least to an untrained eye) essentially indistinguishable from a true random source.
Your parallel comment seems to indicate that there isn't a lot of great practical reading on this topic; I don't suppose you've seen a discussion anywhere going through anything like the approaches above, and whether there is a way to do it "properly"?
I've been meaning to write some kind of public exploration on this stuff so people can tell me some other source explains it better, but haven't made the time to do more than write code like the blake2b demo I had handy to spit out numbers.
For example, if you were to run Blake2b in 8 bit mode with the above construction, what would your PRNG period actually be? In the ideal case it would 2^(8). In practice, the best you'll get is 26 or 10% of the ideal case. There's a significant probability you won't get that either for a given seed, as much of the state space actually drops into a lower period cycle. The issue is that a real hash isn't the correct construction here, even though a mathematically ideal hash would be fine.
I'm not aware of any actual reference material for that kind of stuff and the people I've met who are knowledgeable learned from seeing other's mistakes.