What are the 2008 honda odyssey alternator problems?

The biggest problem with alternators on a 2008 Honda Odyssey is when they fail and no longer charge the battery. Of course, this isn't particular to this vehicle, and will eventually happen to any. You'll want to verify it actually is the alternator before you go buying a new one. To do so, start the engine, and touch the red and black leads of your meter to the battery; if you reverse the polarity, it'll just give a negative value, which is fine. If your alternator is doing its job, you should see around 13.5 volts. If you only see the battery voltage, it's likely time to replace the alternator. Cost for that is going to usually be around $200 for labor and up to $700 for parts. Replacing an alternator is somewhat labor intensive, what with having to deal with the timing belt, but it's certainly something an enterprising DIY home mechanic can tackle.