I was thinking that if I were to take my new baby and my wife out for a hiking/backpacking trip, it might be really cool to have a lightweight rickshaw-like device to tote the baby. It seems to me that would be easier than having my wife carry the baby and me carry all the gear. Has anybody seen anything like this, or should I brush up on my MIG/TIG welding and sewing skills?

I remembered seeing one of these somewhere & took a shot @ bicycle world.com. Resulted in a link to
Chariot Carrier
Some of these can be converted modes Modes

A bike trailer - they often come with a stroller mod.
Just be careful, a baby can't really handle the jostling of a trail.

Consider this:
Baby weighs about 10 Lbs.
Rickshaw or other baby bucket device weighs 25 lbs and requires at least one hand to pull/push/carry. Add baby and that's 35 lbs carry.

A baby sling for you or your wife weighs less than a pound and keeps both hands free, keeps the baby warm and bug-free and is MUCH lighter and easier than carrying a 25 lb trailer around.
I used to carry my son in one when hiking, as did my wife. You don't have to get one with "duckies" on it. Ours was a soft denim like fabric. The little guy loved it, we kept our hands free and all was fine.

We used a Baby Jogger to haul our kids around - both on our acreage and on trails. It worked very well. We could even roll up stairs, curbs, and even over couches (I actually tried it). This thing was so versitile we came to refer to it with the military-style acronym ATTWIEV - All Terrain Three Wheeled Infant Excursion Vehicle. It sounded much more macho than "Baby Jogger".

Check out http://www.babyjogger.com (I have absolutely no connection to the company).

Don't be fooled by lower quality imitations. The Baby Jogger is a bit (a lot?) more expensive, but it is built like a tank and easily folds flat for stowage. Our's survived through our two kids and then we loaned it to my friend when he had his first child. It is now going strong for their third child and still going strong for them.