Raise heck with the dealer. Odyssey's are notorious for having bad transmissions. No way it should be failing at 60K miles..
We used to be big Honda fans... then we got our 2002 Odyssey.
I did a quick google search for Odyssey transmission issues and Gordon is 100% correct as always. They have had a ton of issues with those.
http://en.wikipedia.org/wiki/Honda_Odyssey
The five-speed automatic introduced in 2002 was supposed to fix this, but general reliability of the transmission did not improve until 2004. The five-speed introduced in 2002 suffered early wear out and failure of the transmission's third clutch pack. This causes large amounts of clutch debris to migrate through the transmission and block flow of transmission fluid, caused slipping, poor or no shifts, or sudden down-shifts from 5th gear to 2nd gear. Under some conditions, the second gear could overheat and break, causing the transmission to lock. An oil jet was added to lubricate this gear but this did not solve the third clutch problem[2].
http://www.hondaproblems.com/recalls/Odyssey/2002/
NHTSA Recall #04V176000 POWER TRAIN:AUTOMATIC TRANSMISSION
APRIL 15
2004
1,099,796 Odysseys recalled
2002 HONDA ODYSSEY -- On some mini vans, sport utility and passenger vehicles, certain operating conditions can result in heat build-up between the countershaft and secondary shaft second gears in the automatic transmission, eventually leading to gear tooth chipping or gear breakage.
Gear failure could result in transmission lockup, which could result in a crash.
On vehicles with 15,000 miles or less, the dealer will update the transmission with a simple revision to the oil cooler return line to increase lubrication to the second gear. On vehicles with more than 15,000 miles, the dealer will inspect the transmission to identify gears that have already experienced discoloration due to overheating. If discoloration exists, the transmission will be replaced if discoloration is not present, the dealer will perform the revision to the oil cooler return line. The recall began on April 21, 2004, for Pilot, Odyssey, and MDX owners. Owners of the Accord vehicles will start receiving letters on June 28, 2004, and on June 29, 2004, for owners of the TL and cl vehicles. Owners should contact Honda at 1-800-999-1009 or Acura at 1-800-382-2238. For detailed information & supporting documents, see the official NHTSA page concerning recall #04V176000 »
http://www.carcomplaints.com/Honda/Odyssey/2002/
Hope this helps!