What can I tow with my Honda Odyssey?

We have a Toyota Sienna. Our Pop-up is right at the max and tows well, but I wouldn't want to tow a full-height trailer anywhere near the max. We get around 16 MPG towing in FL. The one that you are looking at probably doesn't include any extras such as battery, propane, possibly AC, etc. Realistically, the dry weight before you add your stuff will probably be at or over 3000. With kids especially, it is remarkable the amount of stuff that you "have" to take. The weight will shoot up pretty quickly if you aren't very careful. I'd be looking at Pop-ups if it were me. The biggest disadvantage to Pop-ups is the set-up/tear-down time compared to a TT. Once you get it down though, it isn't too bad. I can go from parked to adult beverage in about 25 minutes if I set everything up. I can make it sleep secure in about 10 minutes if necessary.

You are going to want to make sure whatever you get can use a WDH or you will really need to watch the hitch weight. Mini-vans are designed to move people comfortably and typically have soft rear springs for comfort.