Can I drive a minivan (family van) for Uber? I drive a Honda Odyssey.

"Minivans" are ok. Big vans, box trucks, and similar are not.

Yes, you can drive minivans such as Toyota Sienna and Honda Odyssey for Uber.  In fact, you would qualify for both UberX (4 passengers) and UberXL (6 passengers).

Minivans are becoming more popular amongst drivers, as they can earn more money by being classified as UberXL and they offer great gas mileage.

We have heard the confusion before.  On Uber's vehicle requirement page, they state, "No vans, box trucks or similar."  By this, they mean large vans like this NV400 that are often used as commercial vehicles:

With that said, smaller NV200 is classified as a minivan (or "NV Passenger"), so it is permitted. 

Uber published a list of vehicles for NYC, this may be helpful.

List of Eligible Vehices - New York City

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Uber Driver, CEO of TaxiFareFinder and Unleashed, LLC