Lightning strikes can cause damage to many items in a house. The most suspectable items are GFI outlets, any electronic machines ( TV, computers, audio equipment etc.), dimmers, switches and elements of an electric range/oven, and yes, garage door openers.
Another area to check is to unplug any device plugged into a receptacle and inspect the metal plug prongs for any signs of burns. If you find dark pits, any melted metal etc, then the plug and outlet should be replaced.
Using an inexpensive outlet checker, check all outlets for malfunctions such as open grounds, neutrals etc. Open the circuit breaker box and do a visual check for any signs of arcing or burned wires.
If you are not comfortable doing these inspections yourself, ask your electrician to do a quick once over on the items I mentioned above. This should take less than an hour to complete and give you peace of mind.
Just a side note. Lightning rod systems are somewhat effective for direct hits, however in the large percentage of lightning damage cases, the surge of ultra high voltage enters the home via the grounding system of the house. Lightning strikes a tree, a transformer or the ground close to the house and energizes the entire ground and neutral system and conductors. The instantaneous voltage spike can easily exceed 50,000 volts. there is little you can do to defend your property from such a strike. Common surge protectors will fail most of the time. Computers connected to a UPS usually survive because they are isolated from the primary power source even though the UPS may be damaged.
Having done a LOT of this over the years, your options are basically - find something else to follow, go plenum-rated and use ductwork if you have it as @Comintern suggests in a comment, run wires on the surface or go though a lot of agony trying not to open up walls. In many cases, opening up walls would be a lot less agony. There are often inconvenient things like firestops in the middle of walls, and you can get into all sorts of excitement you'd just as soon not when trying to drill blind or around corners or any sort of fancy work like that. Yes, long flexible drills are made - without a long flexible camera to be sure what the heck one is drilling into, you can get into a world of misery trying to use one...
My prime suggestion - open the walls, run conduit, never have to do this again (if you run the conduit right so that you can pull these wires out when you need to install whatever everything is using in 10 years.) In large part you may be able to minimize opening first floor walls if you can distribute across the basement and have short runs up to the first floor. The run from basement to attic will probably require it - you can distribute much of the second floor through the attic to minimize opening walls up there.
If you can't use plenum rated cables in air ducts (and DON'T use non-plenum rated cables in air ducts, since you and your family are the ones that will be subject to smoke inhalation for the savings of a few pennies) look around the plumbing for possible routes you might get a network wire in. However, most non-owner-built homes have miserable access to the plumbing, so that may not save you much if any wall opening. Builders who are not going to live there seem to love just sealing it all up in drywall to make more work when service is required...
Look around for things like stacked closets to try and locate space where you can open the walls with minimum area to repair/repaint to run your conduit or wires.
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My understanding of lightning rods is that the general design is just as you describe, but you must contact the local electrical code enforcement for the exact specifications, equipment, and procedures. They may give you advice on how you can install this yourself or they may require you to have it professionally installed. After all you don't want to have happen to your house what happened to Mark Twain's house in NY. Mark Twain's lightning rod story
BTW it is my understanding that Twain's story is without any scientific basis even though it is delightful tale. Lightning rods actually don't attract lightning, but instead prevent it from striking in the first place. The idea is the sharp point on the rod discharges the charge in the earth induced by a charged cloud overhead.