Half the work is saying what the problem is
Kettering's starter motor came from restating the problem. Some problems can only be stated after trying to solve them.
A problem well stated is a problem half solved.
Starting a car in 1910 meant turning a hand crank at the front of the engine, and a crank that kicked back could break an arm. An electric starter was the obvious fix, and the obvious objection was size: a motor strong enough to turn over an engine was assumed to be far too heavy to carry around. Charles Kettering, who had built small electric motors for cash registers at National Cash Register, stated the problem differently. The motor only had to run for a few seconds at a time. A small motor could be driven far past its continuous rating for that long without harm, and with reduction gearing it could turn the engine. Cadillac ordered twelve thousand of his starters and fitted them to its 1912 models.
That story is the best argument for the saying, and it is also why the saying attaches so easily to Kettering. There is, however, no document in which he is shown saying it. John Dewey wrote in Logic: The Theory of Inquiry (1938) that a problem well put is half solved and described it there as a familiar saying, so the idea was already in circulation and neither man can be named as its source.
In practice the advice is about the sentence a team writes before it starts building. “Improve onboarding” is a direction. “New users leave at step three because the bank connection fails for two of the largest banks” is a problem, and most of the solution is visible inside it. The second version took more work to write, which is the point: the stating was half of the work.
The saying fails for a class of problems that only reveal themselves through attempts. Horst Rittel and Melvin Webber called them wicked problems in 1973: problems where each attempted solution changes the understanding of what the problem is. Most new products live partly in that territory. A founder cannot fully state what customers need before showing them something, and the clearest problem statements often come after the first version has failed in a specific way.
The two ideas fit together once the order is right. State the problem as precisely as current knowledge allows, build against that statement, and rewrite it when the evidence says it was wrong.