I am designing a web site in which users solve puzzles as quickly as they can. JavaScript is used to time each puzzle, and the number of milliseconds is sent to the server via A
Even a compiled application could be forged. If the user changes their system clock halfway through timing, your application will report an incorrect time to the server. The only way to get an accurate upper-bound on the time it takes them is to start timing on the server when the puzzle is given to them, and to stop timing when they supply the answer.
As others have pointed out you can minimise the effect that slow connections have by making the load of the puzzle as small as possible. Load the entire page and "game engine" first, and then use an asynchronous request to load the puzzle itself (which should be a small amount of data) to level the playing field as much as possible.
Unfortunately you can't do latency compensation as this would be open to tampering. However, on a connection that's not being used for anything else, the latency for a request like this would be greatly overshadowed by the time it takes a human to solve a puzzle, I don't think it will be a big deal.
(Reasoning: 200ms is considered very bad lag, and that's the average human reaction time. The shortest possible "puzzle" for a human to complete would be a visual reaction speed test, in which case bad lag would have a 100% markup on their results. So as a timing solution this is 2-OPT. Any puzzle more complex will be impacted less by lag.)
I would also put a banner on the page saying to not use the internet connection for anything else while playing for the best possible speeds, possibly linking to a speed / latency tester.