I have a few sites hosted with GoDaddy. The deluxe hosting plan allows me to have a single domain and unlimited subdomains. I have a Drupal site on one of the subdomains. On the account activation email, I would like the link to the one-time URL to be http://www.name-of-subdomain.com/....
Instead, the link in the new account's login URL reads: http://www.domain.com/name-of-subdomain/...

How can I edit the [user:one-time-login-url] token so that it does not include the parent domain name in the URL link?



Difficult. You could try setting $base_url in settings.php then clearing caches. That might force the one time login link to work correctly.

This system many shared hosts (especially using cPanel) have for creating 'addon' sites is a pain. You could try looking at the niche Drupal specialist hosts, who probably use Virtualmin control panel and avoid this type of setup (though I am not certain they do). Virtualmin is less intuitive but more rational (if that makes any sense). In addition these sepcialists like Drupion and Civihosting and Hotdrupal will almost certainly perform much better have other advantages for Drupal.