Replacing one WordPress site with another one.

About to develop a new site for a client, who currently has a very basic WordPress site running. Trying to determine the best place to develop a new site, while keeping the existing site running.

Options I’m considering are developing it in a subdirectory on the clients server, or developing it in a subdirectory on my server. Just want to know which will be less problematic when it come time to replacing his current site with the new one. Can’t seem to find many tutorials for replacing one WordPress site with a new one.