Pathfinder Unchained describes Lore as
You possess a specialized area of knowledge, generally narrower than that of a full-fledged scholar.
and
A Lore skill must be narrow—far narrower than the most relevant Knowledge skill. The broader the scope of a given category of Lore, the shallower your knowledge is on that topic. If you know about taverns in a wide region, you know less about each of them than you would if you had Lore in taverns of a specific city.
To me that means that Lore(elven history) should give you narrower but deeper expertise than Knowledge(history). A question about elven history should be easier to answer with the former than with the latter, and a high roll should give more tidbits of information.
But the rules don't seem to work that way.
Lore skills use the same DC scale as Knowledge skills: DC 10 to answer easy questions, DC 15 for basic questions, and DC 20 to 30 for really tough questions. In many cases, Lore can substitute for a Knowledge skill, such as Lore (elven history) filling in for Knowledge (history) in a check involving elves.
There doesn't seem to be a provision for specifying that the difficulty of a question depends on what you're using to answer it.
The particular case that prompted this is a PC wanting to max out Knowledge(local) because it's useful and also to take Lore(storm giants) because he has storm giant ancestry, but I'm interested in the general case.
How do I finagle things to get what I want? Right now the best idea I have is to multiply a Lore check by some factor set when the skill is taken that reflects how specific it is, defaulting to two. That effectively turns "really tough" questions (DC 20-30) into easy-to-basic questions (DC 10-15). It also means that while getting a 30 on Knowledge(local) to get the abilities and weaknesses of a storm giant (CR 13) gets two pieces of info, the same result on Lore(storm giants) gets at least eight. I'm not sure whether that's the appropriate amount.
How have other people dealt with this?
I'm going to preface this by saying that I haven't really had a lot of occasion to use the Lore skill a lot.
The way Knowledge and by extension Lore seem to work is that they rely on the GM to determine what a basic, easy, etc. question are. You should expect a lot of table variation, because the skills do not have a lot of examples for what constitutes different levels of a check.
I don't feel lore should be a straight out multiplier. In most cases, a GM should be able to just keep in mind that Knowledge is for generalists and Lore is for specialists. If you have a piece of info that is about elven history, it will be an easier question for a specialist to answer. The actual mechanical benefit of this would be to effectively reduce many DCs by 5 to 10 points.
An extended example: In Golarion, many questions related to the Runelords have high DCs. The DC to identify a specific Runelord by name might be a hard question (of DC 20 or 30). A character with Lore (runelords) likely knows the nature of the runelords as an easy answer and a specific name is pretty basic (DC 15).
To address your specific example of storm giants, I would not have lore be a multiplier for knowledge checks, it sets a new baseline. The baseline for Knowledge (local) to ID a storm giant is CR 23 and you get an additional piece of info beyond that. ID'ing a storm giant with Lore (storm giants) should be, you know, a really basic task (I would say DC 5 or 10, the kind of thing you could do on sight). So you would know an additional piece of monster lore for every 5 point after that. So mixing our examples, a check of 30 with Lore (storm giants) at DC 10 would give 5 pieces of information.
EDIT because I re-read what I wrote and oh boy could it use some cleaning up.
Also, one more note as an aside and to be a killjoy, but the Lore skill description offers "Drow" as an example of not being specific enough to merit being a Lore skill option, and offers "Drow Matriarchs". By that logic, specific races may not be specific enough and your player may want "Storm Giant Leaders". Then again, after double checking, "Frost Giants" are acceptable and "Giants" aren't. That's useful.
It sounds like you're the GM and I absolutely encourage you to ignore that (I think it's dumb myself), but I felt like it was worth pointing out.
Also, one more note as an aside and to be a killjoy, but the Lore skill description offers "Drow" as an example of not being specific enough to merit being a Lore skill option, and offers "Drow Matriarchs". By that logic, specific races may not be specific enough and your player may want "Storm Giant Leaders". Then again, after double checking, "Frost Giants" are acceptable and "Giants" aren't. That's useful.
It sounds like you're the GM and I absolutely encourage you to ignore that (I think it's dumb myself), but I felt like it was worth pointing out.
<nod> I saw both of those and decided (yes I am the GM) that "frost giant" was the more applicable precedent. I think the difference is that drow pop up a heck of a lot more so Lore(drow) is, in effect, far more broad than Lore(_____ giants) even though each is a single race.
I understand what you've said about new baselines. It looks like you'd reduce the DC by 5-15 depending on the original 10-30 DC, which I suspect might well work out in practice to "cut the DC in half."
I might argue that "easy questions", "basic questions", etc. are relative terms. What might be a "tough" question to someone with a broad Knowledge (local) skill might be an easy question with someone with a Lore (storm giant) skill.
You could also give much more in-depth knowledge for passing the Lore check. For example, in combat the same Knowledge check that gives them one piece of info could get them 3+ pieces of info, plus some of the ecological information about the creature. Under the concept that passing a Knowledge check means "you've heard about this once" whereas passing a Lore check means "you've read a book about this."