Is there a technical word to reference the total time a user spends on a website in one session? Something like "Time-on-Site (TOS)"?
As Jaslr's, Chris' and Todd's answers point out, the best word differs in different contexts. Answer these questions to help us sharpen our answers: What are you measuring, exactly? Articulating this may point to the best term. Why are you measuring that? What's your larger purpose? Again, looking for the root purpose, meaning or intention behind your question. Who is your audience when using this term? Unless your case is to teach, figure out what term would be most clear to your audience.
So, no, not really. Observing and measuring users is not scientific, not precise and so there is no single convention. If I had to pick one, "session" because its plain and in common use amoung developers and usability pros.
It's often called a 'session' or 'session length', though the exact meaning can differ depending on who you talk to.
In some cases a session will refer to the period of time before a person has to sign in again, where for other people it lasts as long as someone has the window open and is on the site, and that a new session starts as soon as you leave/close the window and come back.
If in doubt, 'time on site' or 'length of visit' seem very clear to me and won't get you into trouble.
Apart form the suggestions made so far, one term that I like is
duration of visit or
length of visit.
To me, it feels less technical and more something that actual humans do - visit websites.
If you have a specific audience, use the term they are familiar with, though. For people familiar with google analytics, that would be
Visit Duration, and
Avg. Visit Duration respectively.