You can't replace the function, but you can substitute it:
Copy the function to your child theme's functions file.
Rename the function.
Modify it as needed.
Find the template file that references it, such as header.php, and copy that to your child theme.
In your child theme's new template file replace the reference to the original function with your renamed ...
get_next_posts_link() and get_previous_posts_link() are for linking between pages of a paginated post archive.
For navigation between individual posts you want get_next_post_link() and get_previous_post_link(). Note the singular "post".
echo get_next_post_link( __( 'Previous post' ) ) . ' | ' . ...