This is what I could find in 30 minutes on Google Scholar.
Lots of papers on the entropy of human movement and walking gait. Lots of papers on measuring the entropy of passwords (for the purposes of displaying a strength meter). Some papers talking about the amount of entropy a human can memorize.
This paper is pretty close and probably contains some partial answers:
If I am reading Table 1 correctly, then they found that when allowed to make 8 - 10 character passwords, people tend to select passwords from a roughly 240 password space. This is kinda a measure of entropy, I guess.
Here are other papers I found that are close-ish to the spirit of your question:
So within the powers of 30 minutes of my Google-Fu, I am going to say Lots of research has been done on the average entropy of passwords found in database leaks (hint: it's real low).
But as to "what's the best a human brain can do?" It looks like: No, this research has not been done.