I am using a phone, so i don't know if it will work the same on a computer (although it should). When you create a new meme and upload your own image, once you click on the share box, an empty "Name" field should appear below. That is where you can type in the name that you want. That will also become a tag in the template. Once you ok the upload it should take you to the page to make your meme. It will be on that page where you can customize the position of the text boxes and add more.
As far as changing the name once the image is already uploaded, i wasn't able to figure that out. Someone else might know, though. Otherwise you might have to upload the image again to add the name.