Yeah, generally you'd need root access to modify the "/home" directory, which includes renaming items within it, or creating new items such as a symlink.

With root, however, there should be no problem at all with it, although I'd probably use cp -a /home/user /media/mmc1/user, rather than -R as this will preserve symlinks, modification times etc.

By default, the MMC card is formatted as FAT - a DOS file format - meaning it can't hold all the semantics of the Unix file system which /home/user is within. If it's a problem for you, you can relatively easily reformat the MMC to a "proper" Unix FS.