Honestly, it would be great for them to add a unique ID. I've yet to see that they have. However, you could create your own user-base system first. The user should create an account on your site, and then add their tumblr to it. This way, when they log in and url's have changed, etc, you check for it and update your current user, this goes for so many other assets too, followers, posts. Yadda.
Some comments on this post are pretty old, but i've easily saved their token/secret into the database from the first time logging in. Granted, it will change during the next log-in.
My biggest issue is adding their followers to the database. If I track a user's followers, (aka by their url/uid), and they eventually change it, and someone else grabs that uid, then they could easily take place of the other user...
So a good way to fix this issue is to run a test. (This is the only best possible solution I could think of.) That before someone creates/connects an account it can test/compare their most recent post, with the uids most recent post, thus proving if they are, or are not, the user in question.