First, understand that the homeowner isn't liable for this unless the homeowner was negligent in some fashion. How was the homeowner negligent in the tree limb snapping? If there were obvious signs the tree limb was rotten or in otherwise poor condition, your friend's husband has the problem that it should have been apparent to him too and he should not have relied upon it. In that case his own contributory negligence might kill any claim he had. if the limb wasn't bad (or the limb's weakness wasn't something the homeowner could have known about) then I don't see any negligence on the part of the homeowner.
Following on that, most homeowner's liability policies will have language something like: “If a claim is made or a suit is brought against an insured for damages
because of bodily injury or property damage caused by an occurrence to
which this coverage applies, we will . . . pay up to our limit of
liability for the damages for which the insured is legally liable.” Note that it doesn't cover all injuries that occur on the property; that's something that a lot of people don't understand. Instead, it just covers the homeowner for injuries on his property for which the homeowner would be legally liable, e.g. for which he could be sued for negligence. If there is no legal claim against the homeowner, the homeowner's insurer won't be obligated to pay the claim. Thus, even if there wasn't a statute of limitation (SOL) issue here, there may have been nothing to get from the homeowner's policy.
Second, and to the point of your question, the answer on the SOL depends on the law of the state where the home is located. If it was Ohio, then Ohio Revised Code section 2305.10 provides the SOL for personal injury claims. For claims other than product liability and sexual abuse claims, the SOL is two years from date of the injury. Thus, it appears that even if he might have had a good claim, he's out of luck in Ohio.