If you're able to directly discuss what her plans are for after death (cremation etc) then could you just talk about your own plans in the same context - don't explicitly suggest that she get it done, just discuss that it's what you want for yourself.
This is a good idea. Mentioning up front that you think it's a long shot makes you sound a lot less crazy. Definitely try to raise a discussion, instead of lecturing. It comes of less as 'Hi, let me induct you into our crazy cult.'
Ask her about her reasons for her wishes for disposal of her remains, and then come to terms with the fact that her reasons are (probably) not the same as your reasons.
On more practical grounds... is the money to pay for the cryopreservation available? Cryonics isn't impossibly expensive, but it's not cheap.
She's a woman, so whatever difficulty you are expecting, double or triple it. Women don't like cryonics.
She is not particularly religious, but is concerned with leaving as much money for my grandfather (and later my parents and me) as possible.
Yeah, I'd give up here. Signing up is hard, it's expensive, it's much too late, there's a sure-fire competing desire, and the target is female. The odds of success are, at a minimum, <5% (if you actually try, I'd be happy to record a prediction or bet on it). This will not end well for you. Don't try.
The odds of success are, at a minimum, <5%... Don't try.
One does not necessarily follow from the other. Don't expect too much (I would put the odds around 1% myself) but it might still be worth a few hours or days of your time.
You can look at cryonic signup rates by gender, and there's also the article that advancedatheist linked. I'll add that in my anecdotal experience, women seem more likely to dismiss the idea when I bring it up in casual conversation.
For myself, personally, I don't like cryonics because I think the research largely points to it being non-viable. Of the three other women I can remember speaking to recently, two others had the same objection, and the last one's issue is that they live in Australia so the difficulty of getting cryopreserved soon after death is ridiculously high (they view it as plausible-but-unlikely)
Would people set a higher odds of success cutoff for starting a cryonics argument than they would for an extension of life treatment?
I would. Advocating cryonics, a known minority belief, has the chance to blowback and make you look like an ambulance-chaser or worse. Even if the dying person would calculate that cryonics is +EV for them, that doesn't mean that it's a good idea for someone else to try to intervene and get them to sign up (unless that other person is perfectly altruistic and doesn't mind the possible negatives they might suffer).
Advocating some horribly painful low-value - yet medically approved - treatment or heroic measure, on the other hand, has no downsides for you.
I'm not clear on whether I should advocate this, but I wonder if you could spin not-cryo as a conspiracy (without outright lying):
"Have you heard of cryonics?"
"Heard of what?
"Yeah, didn't think so. They have a hell of a time getting past the typical story about death. Cryonics isn't even like crackpot theories, like the Rapture or what have you, that get a hearing simply for being ridiculous -"
"Okay, but what are you talking about?"
"There are organizations that will preserve legally dead bodies, frozen. The defini...
New icebreaker & rebuttals:
"If there was a technology that might save your life, would you consider it?"
This is going to get an almost definite yes.
If no: Ask her "So, you WANT to die?" - this is likely to snap her out of whatever fog she's in if she's still got a significant will to live. Ask that here. If she shows enough will to live, ask the first question again, you may be able to continue. If she actually wants to die, which is not inconceivable for a person in pain, obviously stop here, determine why, and talk it out. ...
Ideas:
"There is new information in the field of cryonics, do you mind if I make sure you know the important pieces of info?"
(That gets you space to talk about whatever technological advancements you want to talk about or whatever, get her thinking about it, and make sure she's informed.)
If refused: "From my point of view, this is something that could save your life. If it can save your life, then no matter how small the chance is, I'm ethically obligated to talk about it with you, right?" (Make sure to phrase it as a question.)
(Even ...
Heavy use of dark arts, not logic or rationality is your best, if flimsy, hope.
You might be able to sell it as a really expensive burial of his wife to the conspiracist grandpa, and to both of them together as a glimmer of hope of them reuniting in the after(cryo)life. You might be able to use what Alicorn suggested on your grandfather, if you spin it right. Think about other arguments they (not you) might find convincing enough to spend a few hundred thousand dollars on. Is either of them big into lottery or gambling? What other biases that can be exploi...
I have found that though the dying and the imminently bereaved won't go for it, they are nevertheless profoundly appreciative of the thought.
My paternal grandmother is dying of cancer (not brain cancer). She is still relatively healthy, and is taking chemo, but there is little hope of remission (and even if that does happen, she'll probably die of heart failure fairly soon). Her current plan is to be cremated and have the ashes buried in a graveyard (in my opinion, the worst of both of the "standard" approaches, but that's not the point of this post).
I would prefer if she were cryopreserved, but am unsure how to even begin to broach the subject. I also have no idea how to convince her. She is not particularly religious, but is concerned with leaving as much money for my grandfather (and later my parents and me) as possible. I have previously discussed cryonics with my parents; my father brushed off the idea and my mom looked into it but dismissed the idea because the future isn't likely to want her (I find this argument ridiculous on several grounds). This means that I can't count on them to help talk to my grandmother. I may be able to talk to my grandfather first, but this would probably not be much of an asset: he is into several different conspiracy theories (the most recent ones center around the world secretly being controlled by the "elites" who use the U.S. President, U.K. Prime Minister, etc. as figurative puppets), but my grandmother doesn't seem to believe these and probably wouldn't listen much to his talk of cryonics either.
Any suggestions of how to broach the topic or convince her once the topic is broached would be appreciated. I am currently at my grandparents' house, but am leaving less than a day after posting this (most of which will be spent at the local nighttime, and thus asleep). I would prefer not to upset her, both for obvious reasons and because I may not be able to bring myself to bring it up on the day we depart if it will cause us to leave on a bad note.