I'll take a "crack" at it. All lubricating oils begin life as crude in the ground. It is pumped out of the ground, shipped to a refinery where it undergoes certain processing. Each step adds cost, but allows more carefully controlled properties for lubricating.
Crude oil is made up of 'polar' components that contain nitrogen and sulfur atoms, and hydrocarbons that contain only hydrogen and carbon. The first processing step is separation of the polar parts from the hydrocarbon oils in a step often referred to as solvent extraction. This gives the least expensive type of oil used in lubricating. (API group I)
Just as with fats that we eat, the solvent extracted hydrocarbons contains chemicals called "saturates" and "unsaturates". (The difference is that with lube oil it is the saturated compounds that are the most desirable.) The second processing step involves "hydrogenation" (just like with edible fats), in which extra hydrogen is added to the molecules to make them more "saturated". This second step can be done several different ways- hydrofinishing, hydroisomerization, hydrocracking, or severe hydrotreatment, which produces progressively better oil at progressively higher cost. (API Groups II and III) Note that all of these categories could be referred to as 'hydro-treatment', so you don't know where on the scale the quick lube's oil falls.
A third step can be done, loosely called synthesis, in which the lube oil molecules are broken down to simple building blocks and they are then put back together to have the best properties available, but this process is the most elaborate and expensive. (API groups IV and V)
What the quicklube is trying to imply is that they offer a better oil than the most basic, yet not at the price of a full synthetic. The catch is that with specifications going up every few years many oil manufacturers are having to upgrade from Group I basestocks just to meet them, and so without more information it is hard to know if the quick lube is making a distinction without a difference. Ask the manager whether they are using API Group II or Group III basestocks and see if he/she gives you a blank look- if they do I wouldn't read much signifigance in their hydro-treated claim.