I have installed several ground rods over the years (related to antenna installations). In each case, I've taken what some may erroneously consider an extra step, tying (bonding) the new rod to the existing electrical service grounding system, using a #6 AWG copper conductor. My reasoning was similar to yours, there was a significant distance between the antenna mast and existing electrical service grounding system... I wanted to provide access to earth as soon as practical. However, I also wanted to have some assurance that the antenna system was connected to the same grounding system as the electrical service, to avoid a big voltage difference between the antenna system and any other grounded devices in the house.
The NEC calls for a minimum conductor size and choosing to use a heavier conductor is certainly a choice that errs on the side of caution. If the question is whether to build a system so strong that it can't be broken, consider... As you have already observed, a direct strike could easily cause failure of light gauge conductors. If the weakest link in the grounding path of a lighting strike is the coax, one can have a reasonable expectation that the coax would fuse open first (provided the coax is connected to a substantial grounding system, outside the building) isolating the outdoor portion of the system from the cabling inside the house. If the mast grounding conductor failed next, you could then have some hope that with the coax already open, there is a significantly lower chance of the energy entering the building (at least not through a failed coax). I propose that a #10 AWG copper wire is going to have a higher fusing point than RG-6 coax and so it's an appropriate size for protecting a receiving antenna system. (But one can still argue that bigger is better... if cost is not a factor.)
The bottom line for me is, I want to take reasonable steps to encourage fault current to stay outside my house. I also want to take reasonable steps to prevent a significant voltage difference from developing between to objects in my home if they are easy to come in contact with.
Also consider that the impedance of a driven or buried ground rod can be surprisingly high. The last two I drove were 5/8" X 8' copper clad, into sandy glacial till, common to the NW. I calculated the effective impedance of each one at approximately 1500Ω. (Based on measured AC current flow and known voltage applied. Clearly, it would take the voltage of a lighting strike to make any significant amount of current flow. I don't depend on the soil to be a conductor that's capable of carrying fault current while holding voltage differences low enough to ensure safety. That's the job of the #6 AWG copper that ties (bonds) the various parts of my grounding electrode system together. If you choose to add a rod, please be sure to bond it to the existing electrical service grounding system.
Take a look at the options you have when it comes to mounting hardware,
http://www.3starinc.com/antenna_mounts_and_masts
FWIW: My current antenna installation is mounted on a 5' Ronard tripod which holds a 10' 6" chain-link top rail tube (Home Depot). The lag bolts are run into 2X4 bracing block glued and screwed to the underside of the roof sheeting, between rafters. The base clips of the tripod and the anchor bolts are sealed to the fiberglass/asphalt shingled roof with mastic tape (Scotch 2228) and roofing cement (Henrys # 209). My previous installation used a set of wall brackets to support the same type of mast, a 10' 6" chain-link top rail. In both cases, I ran #10 AWG copper from the mast to the nearest ground rod, using corrosion resistant grounding clamps to make the connection to the mast and ground rod.