http://www.cbsnews.com/news/pediatricians-warn-against-tr...
The American Academy of Pediatrics strongly advises against home trampolines. Period. Google "trampoline injuries children" and you'll see why your husband is right. Not only will you have potential for injury and the cost of the trampoline, you will have the cost of your increase in homeowners' insurance.
This is from an insurance web site: "Trampolines have also become what are known as an “attractive nuisance”; attractive to guests entering your property but a “nuisance” should any physical harm come to those using it. According to the Insurance Information Institute (I.I.I.), trampolines are responsible for an average of $300 million a year in insurance, legal and medical costs in this country." The article goes on to add that some insurers will simply refuse to cover you for anything that happens on the trampoline, including any injuries that a neighbor kid incurs. Putting in a trampoline and just not telling your insurer could void your homeowner's policy.
If a stranger gets on it in the night and is injured, you will be held liable, fence or no fence, net or no net, in most places. Even if that stranger didn't have permission to be on your property -- in some places that does not matter and the property owner could still be legally liable for injuries.
Not sure why you are so very sold on this except that it seems to provide a lot of high-energy action in the small space of your yard. Please reconsider. Your husband is right on this one. Even if it's the boys' only gift, it will suck up a ton more money than the trampoline itself costs, and you will always have to worry that if someone enters your yard to use it without your knowledge, you will be held liable. Really not worth the expense and worry, especially since the kids will tire of it eventually.