Before opening this thread, after just seeing the title, I wanted to answer: at least 400-500 man*hours. And I still do. As previous posters said, costs depend on how much time you have to prototype and test and think about the designs before cutting. My observations are that many good robots start bulky at the beginning of the season and get leaner as they improve.
First, if your program director wants to save costs on the new parts, he has to invest space and money into providing extra storage space and organizers to sort all the old parts and have them easily accessible.
Also, to save money on materials without compromising quality of the results, he needs to give students more meeting time and support in the form of mentor's availability and good equipment that makes it easy to refurbish and reuse old parts.
If you already have parts from the previous seasons, I would say you could build a good robot with as little as $600-$800, replacing only what is wearing out and not usable anymore.
That would be 2 or 3 packs of aluminium c-channels, some gears, axles, bearings, washers, collars, spacers. Potentiometers and limit switches are sensors that keep breaking and need to be replaced regularly. That would also include at least one new/spare battery because they decay over time. Also, money invested into giving them more time to practice and go to and extra local competition is well spent and will save some dead-end design solutions, by letting them see other robots and exposing to new ideas.
Again, you can avoid buying new parts if you pay for it by additional student's time.
My team spends first couple of meetings of the season by refurbishing old motors. We didn't buy any new motors this season and were able to qualify for states with 2014-2015 motors, then went to states with 2015-2016 motors and did there very well.
Similarly, we didn't use new potentiometers until end of December, reusing the old ones that still had at least some of the working range. Same goes for HS gears that have few bad teeth. This all takes time to test and repair when possible. Does your program provide students with enough meeting time for that?
The only way this will work, if your director understands that the goal is not to optimize number of trophies per every dollar spent on the program, but to maximize amount of good learning students get within reasonable costs.
It is well known that creativity increases when resources are limited, but if the resources are way too limited then you cannot do or learn anything and it is just a source of frustration and a waste of everybody's time.