What is the cheapest way to get from Zurich to Nice?

The cheapest way is probably ride sharing ("covoiturage") or hitchhiking. But these options require some luck and flexibility.

If you want something more predictable, take the plane or the train.

For the flights, use an engine like Kayak or Matrix Airfare Search to check the possibilities. For the beginning of September I have found one-way flights for +- 110 EUR (Germanwings, Air Berlin, Iberia).

For the trains, you can use Voyages SNCF. I have e.g. found a ticket for 126 EUR by night train via Paris. This has the advantage that you save one accommodation. The price includes the ride plus the bed. On a daytime train you will have to foresee more or less the same budget.

You can also opt for multimodal transport. Easyjet flies from Nice into Basel and Geneva, from where you can easily proceed by train to Zürich. From Basel it is one hour and costs 32 CHF. From Geneva, it is a bit less than 3 hours and will cost 84 CHF. The price for the flight depends on the exact date of travel and the extras you want to take.

The cheapest way is probably a bus. Do compare prices with flights however, sometimes cheap flights are available if you book long enough in advance. Buses are also cheaper when you book in advance but they tend to cap much lower.

Rome2rio is convenient to get a review of the different possibilities. Make sure you check actual prices by starting the booking process, their price ranges are not always very reliable.

At least in summer 2016, Eurolines runs a daily bus from Zürich to Nice, leaving early morning and arriving mid-afternoon. There's also a connection via Turin, but that requires spending a night in Turin.

Zürich to Nice isn't particularly convenient. The straight route would go through the mountain and there aren't that many people doing that route. If you're touring the place, it would be a good time for a stop in Turin or Milan.