From a financial perspective, the concept of "cheap instrument" can be a lot more expensive that non musicians expect.
When I was in high school band, the cheap, second hand version of my instrument was $200 to buy, with another $50 for a good cleaning needed. We rented from the district for the first year (giving me a chance to figure out if this was the instrument I wanted to play), then looked for the second hand version to buy, which was cheaper in the long run. A low end new instrument would have run at least $1000.
From a kid perspective, the parents should generally have a good idea if a child is serious about a hobby and is likely to stick with it. The kid who flits from enthusiasm to enthusiasm on rapid basis and has a closet full of barely used sports, art and music gear gets a thrift store or rented guitar until they've demonstrated they want to stick with it. A musical kid who tends to stick with the same enjoyments is a different story, and could well benefit by starting with a decent instrument.