My wife's 2006 Odyssey had the same problem today.
I started with a new battery as that was easy to replace and was what I assumed was the problem...wrong. $90, but it was due for a new one anyway.
I would start with the starter relay/solenoid...less than $5.
I'm pulling the battery tomorrow to check all of the wire connections since I will not be able to get the relay until Tuesday.
If need be, I will remove the starter and take it somewhere to be tested so I don't have to drive the van and get stuck somewhere far from home.