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,
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
- 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.
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.
Now he wants to put the honest on Michael to prove Sonny guilt when he had the proof and he threw it away.
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 …