I'll chime in as a member of a family with a 400 year long unbroken naming tradition. (Yep, 400 years.) In our case it is the girls who get the name, and it is, in fact, two names. As the names are quite common, the understood rule is that you are allowed to use linguistic variants, that you can use the name as a first or middle name, but the expectation is that at least one of the two will be used at least once in every generation of every branch as long as there is a girl child. So far as I am aware, in 400 years, no one except one of my sisters has ever chosen to deliberately completely ditch the traditional family name set. (My Dad did it one better and used linguistic variants for all of his three daughters. I have one DD, and yes, she did get one of the names.)
I was discussing this with one of my cousins some years ago, who said that she really wanted to buck the trend, but alas, God heard her and she only has boys.
(that's a joke, you understand).
In families like ours, there is emotional investment in the tradition. We're not pushy sorts, so when my sister did buck tradition, nothing was said except to ask once if the omission was deliberate, but I know that many people did feel a bit sad about it. (DSis chose to name her DD after our mom instead, which was rather an unanswerable argument.)
OP, if you were simply being asked to name a child after a grandfather you do not care for, I'd be very with you on it. However, this is not the case here. It is a longstanding family tradition that goes well beyond that one man, and I will bet anything that your DH does care more about the tradition that he wants to let on. (If he really didn't care at all, or if he disliked the name for childhood reasons, he would be probably be more enthusiastic about ignoring it. Since he isn't, I'd assume that he is trying not to hurt your feelings by hedging. Maybe he is afraid of family reaction, true enough, but usually that is a pattern that plays out in other ways. If he normally doesn't put his parents' feelings before yours, then you can probably safely assume that he is at least a little bit invested in this name. After all, it is his name, too; the odds are that he really doesn't see this choice as naming his son after Granddad, he probably sees this as a question of whether to give your son his own name.)
FWIW, the reason that my Dad's family chose to put the naming tradition on the feminine side is that that way it doesn't end up in competition with the tendency of fathers to want to name their boys after themselves. It's also rather devious that way; by keeping it in the female line, the name set is perpetuated throughout the extended family, even though surnames will vary.
Now then, let me ask this ... how do they tell them apart? Most families that have this kind of tradition also have a tradition of variant nicknames, so I would disagree with the PP who warned that the rest of the family are likely to use the name in spite of your preference; that normally is not the case when there are several living people who carry the name. Most of the time in those cases it is understood that the parents will come up with a variant that everyone in the extended family will use, which allows them to keep straight which cousin "Hank" one is speaking of. Is the "first initial" pattern already taken? He could be
H. Harrison Whoosis, if that will work.