This problem I find comes up several times in your development career. Rather that performing a simple file copy and losing all git history, it is actually quite simple to extract the file or folder with full commit history.
Create the patch file
This step creates a single file with all the commits that touch the file or folder. The commits are ordered correctly so they will apply perfectly.
cd ~/repository/path git log --pretty=email --patch-with-stat --reverse --full-index --binary -- path/to/file_or_folder > /tmp/patch
Perform any path replacements as needed
Sometimes you want to alter the path in which your previous repository worked, before importing into your new repository, this can be done with sed (or vim, or any other editor). Note that with sed your must escape the forward slashes.
sed -i -e 's/deep\/path\/that\/you\/want\/shorter/short\/path/g' /tmp/patch
Apply the commits to the new repository
For this step you will need to be in the root of another git repository, and from there you can apply the commits. This will apply fine as long as you do not have any files in the current repository with the same name.
cd ~/another_repository/path git am < /tmp/patch
Let me know if this helped you, or you have any other git tricks that you have found along the way here.