As far as I know, put the honest on someone doesn't mean anything. I believe it is an eggcorn. I think the correct phrase should be put the onus on someone. From the Cambridge Dictionary,
onus
the responsibility or duty to do something:
The onus is on the administration to come up with a balanced budget.
Here are some examples of the correct usage from Dictionary.com (emphasis mine):
Examples from the Web for onus
Contemporary Examples
- In fact, he would benefit by putting the onus of political pressure back on Bibi.
- Trying to put the onus onto someone else for your own decisions is really cowardly and kind of dishonest.
- FDA spokesperson Siobhan DeLancey says the onus is on companies to provide a safe product.
Some of the examples provided in the OP can be found here, where the eggcorn matter is discussed:
Users of this substitution implicitly construe truthfulness (‘honest’) as a burden (‘onus’). Examples-
Again the honest is on the Hampton’s to prove their assertion, while the opposite may be taken by the Sen.’s spouse …
www.lasvegassun.com/.../state-gop-ensig … defenders/
Now the honest is on you to answer some of the question fellow forumers asked.
forum.lowyat.net/topic/1404245
Also the honest is on you to watch what you eat and drink, but we will give you the tools and advices you need to conquer that.
www.iamfitness.ca/pdf/Tips_for_Success.pdf
Now he wants to put the honest on Michael to prove Sonny guilt when he had the proof and he threw it away.
boards.soapcentral.com/showthread.php?p=11841100
I’d recommend you put the honest on the supplier. You should be able to get a half decent kit in the $800 – $1500 CDN range …
www.cctvforum.com/viewtopic.php?f=15&t=678&view=next