Yes this is a fun thread and there is some good and bad advise being thrown around. The little 3ft antenna doesn't need any grounding for RF purposes, and there is little you can do to improve its dismal performance. You can also not ground it sufficiently for it or your radio to survive a direct lightning hit, that's impossible in a home install.
However, you can ground the antenna and make it dangerous as some people have pointed out and that's what NEC is for. If your going to ground it, do it to code at the minimum and that has little to do with lightning and every thing to do with human safety.
As an example, in the 70s I had a CB antenna on the opposite side of the house from my AC entry panel. My 20ft mast attached to the chimney with Hy-Gain Penetrator on top was attached to a ground rod at the base of the chimney. I had a serious problem of the coax for that antenna having about 90VAC on it compared to the grounded wall outlets in my house. I had to wear gloves or be very careful not to touch the house grounded radio case (Pace 1000B heavily modified) otherwise it would nearly knock me on my a$$.
As a coincidence I was visited back then by the FCC, long story, but when they asked me what radio I was using on the night of my infraction, I pointed at the Pace 1000B since it had less mods than other radios laying around. The FCC field engineer promptly disconnected my antenna cable to hook up his equipment and let out a scream and fell back away from the radio as I forgot about the shock problem with that radio. The guy was really pissed off as you could imagine and I now have fond memories of nearly killing an FCC field engineer. In today's world I would probably get sued.
Getting back to the problem, it turns out my ground rod for that antenna was much closer to my neighbors house and AC entry panel, which was oddly fed from another leg of the power system than my house and at a different ground potential. His AC entry panel ground rod was very close to my antenna ground rod and they were at a much closer potential than my own house AC ground. That created some dangerous voltage between my "grounded" CB antenna and the rest of my house.
Legally my ground rod for the CB antenna should have been bonded to my house AC panel ground via the proper size ground wire, which is at least #6 copper according to code. I didn't know about code then and the cost of that much wire would have been too much for my budget anyway at that time.
The moral of my story is you can create a dangerous shock hazard if you ground the wrong way. Not grounding at all would be a safer option in many cases compared to doing it wrong. Some of the complaints of ridiculous and overdone replies to this thread are not ridiculous, they are real and are industry standards as well as meeting code.
For the OP, its your antenna, your house and your safety, if you don't ground the antenna its your choice and for the type of antenna you are using, grounding may not do anything for you. But if you do ground the antenna at least adhere to NEC, and there was at least one link to a good easy to understand way to meet code.
