meaning of Trump's "stable genius"

The word stable can have many meanings. When something is stable, it's fixed and steady. If you needed advice, you'd probably go to your most stable friend, the one least likely to act crazy or be easily upset.

Whether you're talking about an object or a person, the adjective stable implies reliability and strength. You can describe a government as stable, or a relationship, or a desk. A completely different meaning of stable is the noun "building used for housing horses or other animals." Both senses of the word come from the Latin stabilis, "firm or steadfast."

In our case it is just like with the friend mentioned above.

President Trump chose the word "stable" to modify genius simply to refute the common headline that he is "unstable". Meaning roughly, whatever the media is trying to say he is - he's saying he isn't.

TL;DR "stable genius" is a historic term which means a genius without mental disabilities.

The phrase "stable genius" did not originate with Trump, it is a phrase that has been in limited use since at least 1959, which is the earliest use of the phrase I could find. It appears, for example, in the book Starship Troopers written by Robert Heinlein in 1959 and is so far the earliest use of the term I can personally confirm.

...a neodog is not a talking dog; he is not a dog at all, he is an artificially mutated symbiote derived from dog stock. A neo, a trained Caleb, is about six times as bright as a dog, say about as intelligent as a human moron — except that the comparison is not fair to the neo; a moron is a defective, whereas a neo is a stable genius in his own line of work.

Here we also get some context vital to understanding how the phrase is used. The quote explicitly uses the term stable genius to indicate that he is not a "defective". As we know "defective" was a historical term used to refer to individuals who had mental disabilities of some kind. So it is clear that in common usage a "stable genius" is intended to mean a genius without any mental disabilities. This may seem like an odd thing to say but when you think about it when you consider people like Autistics and people with Aspergers who can often present as a Genius in their own fields but still have some mental difficulties in other ways the use of the term becomes clear.