>>>How can I point my new hard drive to the /var/www/html/uploads directory and use my new hard disk as storage?
Actually you have this backwards. You point the /var directory to the new partition UUID or text label in your fstab. If you use lvm you would use links such as /dev/vgname/lvname where the names are strings you specify.
As the previous poster said your first decision is whether you want to use the whole 1 TB drive for the /var directory or reserve space for other directories in the future. I personally would leave some space unallocated for future use.
>>> Inside fstab you can refer to the partition by device name or by UUID.
There are more options than this. You can specify a filesystem label with either MBR or GPT partitioning. If using GPT partitioning you can specify a text partition label.
https://wiki.archlinux.org/index.php/Fstab
As described in the page above the four options are:
Filesystem text label
Filesystem UUID
GPT partition text label
GPT partition UUID