As I understand it, that vehicle has a tow 'rating' of 3500 pounds.  It
also has a Gross Combined Weight Ratio [GCWR] of 8410 (This is the
maximum amount the loaded vehicle and loaded trailer combined can
weigh).  You owners manual should have these numbers somewhere...

Also, the 'curb' weight of your (empty) vehicle is around 4250 pounds.

Subtract that 4250 from the GCWR of 8410 leaves 4160 pounds maximum you
can load and haul.  Two adults plus three kids would be perhaps 650
pounds, leaving 3510 pounds remaining maximum.  Now take the weight of
all of your gear that you will have in the back of your Odyssey. I'm
talking clothing, food, toys, bikes...lots of stuff.  You might be
surprised, but these will likely be another 300 pounds.  That leaves
3210 as the maximum loaded weight of whatever you will be towing.

There is a general rule that, for safety purposes, one should not tow
something that exceeds 75% of the allowable towing weight.  75% of 3210
pounds is 2633 pounds.  This is the maximum loaded weight of any thing
you should haul.

NOW you can start looking at pop-ups and what they weigh.  Remember to
see if the advertised weight includes things such as a full propane
tank, the air conditioner, the fridge, any other add-ons such as a
canopy, etc.  Often it does not.  Also remember that you WILL be putting
gear in the camper.  We have lots of stuff in there:  cooking utensils,
a coffee pot, a microwave oven, life vests for if we go boating, plates,
silverware, tools, wood blocks to put under the camper stabilizer jacks,
 and on-and-on.  I would guess that out 'stuff' in the camper weighs at
least 200 pounds.

That would limit the trailer unloaded weight - but including all the
add-ons mentioned above, to 2433 pounds.

Yes, you can find pop-ups in this range.

Does your tow vehicle have a factory installed towing package.  On some
vehicles this includes an extra transmission oil cooler, larger
suspension, larger battery, larger wheels and/or larger brakes.  I do
not know about the Odyssey.  If not, you will have to look into at least
adding the transmission cooler (and of course, the hitch and wiring).

You will need a trailer with brakes, and a brake controller for your
Odyssey.

HTH

Without first reading the current replies, I'll chime in...

Hells bells!  You're behind the curve!  We started in a tent, bringing along a
porta-crib, potty chair and two high chairs.  We moved up to a new, BIG popup
when the youngest was 6-1/2-years old.  The oldest - of the three - was 9.

With the exception of attending church, there is no better family activity.

Oops.   :(

Stay home.  Otherwise, motels have mandatory A/C.

Remember:  A popup camper is just a tent on wheels.  A/C is nice but requires
an electrical hookup.  That excludes a LOT of nice venues.

Since you didn't mention the rated towing capacity specifically, try this:  
Due to the tow rating, you would be LEAST POORLY served by a popup.  Any other
towable capable of accommodating your family is probably out-of-the-question.

The maximum tow rating for your vehicle is 3500-lbs.

<http://www.trailerlife.com/downloads/01towingguide.pdf>

To achieve any meaningful margin of safety and capability, the popup would
have to be rather small, probably no bigger than a 10-ft box, perhaps even an
8-9-footer.  Family togetherness will take on a whole, new meaning during
inclement weather.

Folding camping trailers are the entry-level of entry-level RVs.  Quality is
not a word often associated with even new, "high-end" popups.  That said,
Jayco and Starcraft and often considered "top of the line".  Fleetwood
(formerly Coleman) may, or may not, belong in that group.

You would do well to choose a reputable, conveniently located dealership and
purchase from them.  That, to me, is more important than brand, particularly
with popups.

Remember, you asked!  Good luck.  You'll have a BLAST!!

 After reading the other replies I'll add this.

You'll most likely want a 10' pop-up to keep the weight down. Try to
find one that has the normal two beds on the end plus some sort of fold
out bed such as a gaucho (couch that folds out into a bed.) Tables that
drop down and turn into beds are a bit more of a pain in the butt.

Older pop-up are definitely lighter. They keep getting heavier every
year. Used pop-ups can be fantastic deals. I would stay away from any
used Coleman with an ABS (plastic) roof. They are notorious for
expensive roof problems.

Don't forget to have a transmission cooler installed in your van. The
cost is a little over $100 and provides cheap and excellent protection
for your very expensive transmission.

So with your new trailer and all your gear and family you'll be hauling
quite a load. You may want to invest in some sort of weight distribution
system such as a weight distributing hitch. This helps to keep the rear
of your van from sitting too low and affecting your handling and ride
quality. To use such a system you'll most likely need a class 3 receiver
hitch and a trailer that is able to accommodate such a system. Many
trailer manufacturers forbid the use of a weight distributing hitch.

You mentioned that you appreciate quality. I'm sorry to say that pop-ups
just aren't built that well. They are on the very low end of the
spectrum for camping rigs. In my own experience I've found that a used
trailer will show all it's faults readily while brand new campers can
have hidden problems waiting to pop-up. If you decide to buy new your
choice of dealer is at least as important as the model or brand you
choose. Around here the camping season is less than 4 months long. If
your camper's in the shop waiting for repairs for 5 weeks it can really
take a bite out of your summer fun.

I would recommend that you do what we did. We bought an older pop-up
(1987 I think) and used it for a year. After we learned our likes and
dislikes we went ahead and made an informed decision and bought a brand
new model. We've since gone back to an older rig due to quality issues.

Good Luck,
Andrew

Thanks for all the great advice.
I think I am going to replace my van next year and get a Toyota
Highlander Hybrid or one of the many diesels coming out next year.
Diesels have that extra torgue and gas mileage I need.

Today I was looking at the Dodge (Mercedes) Sprinter Van.  It has HUGE
storage capacity and at least 5k lbs tow capacity.
http://www.dodge.com/en/2007/sprinter/models/psngr/index.html

Anyway, that buys me some time to find a pop-up.  I definitely need A/
C.  We will going to some pretty hot places and none of us like to
sweat while trying to sleep.

With my Toyota Sienna van I am currently hauling a Rockwood high-wall
which is as tall as the van itself.  I get about 12 mpg but find it easy
to drive well over the speed limit, so I have to still watch the
throttle.  We added air-lift bags to the rear axle to lift the rear of
the van a bit more off the ground (before, we just had a measly 5" of
clearance!).  If we were to go through it all again we would have the
rear suspensing permanently raised 6" off the ground so we do not have
to worry about the air bags since they are a component capable of
failure and also require a compressor under the hood plus a couple
switches that sometimes grind into my left knee when getting out of the
van if I am not paying attention.  Still, it is a rare thing to own a
van with air lift bags and there is some distinction in that :-)   As
for the Rockwood high-wall, I would say it tows just fine no matter what
the pros might say.  We wanted a high-wall for the nice oven, but it
turns out that we really never use the oven at all.  The microwave is
overkill also, since we only like to camp at very natural California
state parks with hot showers and flush toilets and no hook-ups -- so
even our toilet/shower was overkill.  I say get a basic pop-up that
hauls a bit lighter, stay at state parks with flush toilets and hot
showers, and forget getting anything with a toilet that you would have
to clean up after using anyhow.  The good thing about forfeiting the
toilet is that your popup will then most probably have more seating
area.  The high-wall type trailers haul a bit heavy and the only real
feature that we benefit from is they have an extra high ceiling and a
bit more storage capacity.  Ours cost $14k plus taxes so they run a bit
high of price as well.