I went with a scatter-shoot approach, buying things as I found them and hitting a wide range of styles and price points, even a few things that I wouldn’t usually get just to see how they would age. For me it was a ‘reason’ to get age worthy wines and have fun with the experience as they got older, and if my son ends up sharing then all the better. I didn’t go deep or terribly big $$$ but got probably 4 mixed cases worth and have been opening them on his birthday so down to about a case+ with a few blue chips remaining. He is turning 20 next year and starting to get a bit more appreciation for wine and food in general since a trip over the summer to Montreal for music. I told him on his 21st birthday he could pick one of his wines to share with his professor/mentor who was on the trip with him. Might be BDX, might be Cornas, might be Port or Rioja. Will be memorable for him which is all I can ask.
as others have mentioned, its hard to know how your kids will feel about wine in their 20s. It would have been wasted on me until I was in my 30s at least. riesling will age forever and is likely to be a good gateway drug. cali cab is probably safe-ish, sauternes is fair, and champagne is interesting. some of this is way too early to know, though, as most of these wines are just coming into the winery now and probably very few are even dry or pressed yet.
German riesling
We opened birth year wines on our kid’s birthdays since they were pre-teens (and they are now well into adulthood). At first, they got tiny pours but we gradually increased the amounts as we wanted to reinforce that wine was a natural accompaniment to a meal. While wine may not be their beverage of choice today, they continue to enjoy it when they are with us. We’re now down to very few birth year bottles for each of them and my one regret is I didn’t buy enough Monte Bello.
Burgundy, Piedmont, Northern Rhone, Bordeaux, and Riesling
I’m at the other end of this process. When each my children was born, I bought Champagne to celebrate their births with friends and family. When our son was born in 1984, I acquired a dozen Krug Grand Cuvee with the help of a distributor friend (there was literally no Krug for sale in California at that time) and when my daughter was born in 1987, I bought a dozen 1975 Bollinger R.D.
I kept one bottle from each case. I’m going to open the bottle of Krug soon with my son and his wife to celebrate the birth of their daughter last week. Regardless of how it tastes, the wine will be perfect. As was the '75 Bollinger R.D. we drank to celebrate the birth of my daughter’s daughter last year.
When I was first getting into wine, I asked a guy at a local wine store about the type of wine I might want to put in my new cellar. He asked if I had kids and suggested I buy Vintage port if available in their birth year. My youngest is an '07 so it worked out for him.
The guy suggested I buy 3 bottles. One for my son’s wedding day, one for him to drink upon his retirement, and one to be consumed upon his death. Super weird to think about but I did buy 3 bottles of '07 vintage port.
That’s my 3rd son and he’s pretty angsty and mature beyond his age due to his older brothers. He’s pretty tuned out a lot of the time - parents aren’t too bright and not cool at all. I think he also harbors some resentment that we got a bit tired when he was younger and some of that evidence surfaces from time to time.
Recently, the thing about the vintage port came up for his birth year, while my wife and I were sitting with him at the dinner table. I explained the story from the wine shop above and how he’s the only one of our kids to have it. He was floored. I think it changed his opinion of me for a while and maybe some of that still lingers. He said - why didn’t you ever tell me about it? He’s still 15 and it just never came up.
Sorry for the long story but just pointing out that your efforts will most likely be appreciated and good on you for thinking ahead!
^ Great heartfelt story above. We all stumble through trying to do something sweet for those we care about. Sometimes we nail it. Other times not so much. 
It’s the effort that counts.
The old cliche. Parents are dumb but it’s amazing how much they learn when you are away at college.
Great responses here - super interesting to hear the different opinions between @Matthew_King and @R_Frankel . Matthew’s kids are all about dry reds, and Richard’s think the sweeter the better. But it sounds like the one constant is bubbles …which makes sense!
That said, I’m a huge birth-year wine guy in general. My kids are 2010 and 2012 and i went pretty hard. Since you never know what they’ll be into, i bought across all regions/styles but assumed they would like sweet early on so made sure to go heavy sauternes and mosel, but also loaded up on dry reds in 750ml and large format. Thankfully, i havent been sleeping on sparklers…but this thread has me on WSPro looking to buy more!
And as an aside, i know some peeps think that buying birth-year wines is ridiculous because our kids might not give a shit about wine. I totally get that perspective, but in my mind, worst-case ill drink them myself and cheers to my children, my wife, and the effort that she has put into making sure that their lives are the best that they can possibly be 
It’s wonderful reading the stories on this thread and shows that the real answer is to get something one is passionate about to share with children. If you don’t like Riesling, why buy Riesling for your kids?
My initial answer was from a purely strategic perspective; if you drink fairly widely, red burgundy is the way to go from a financial perspective. But ultimately whatever you open is likely to be special, especially since the kids probably won’t care too much about the quality of the wine anyway when they’re 21. 
Who knows what will be in fashion in 20 years. Hardly any of the Gen Z kids I know drink alcohol at all. They certainly love some other drugs though. I’m not sure how sassafras oil, mushrooms, and weed age though. Have a feeling alcohol will go the way of tobacco soon enough.
For my son I bought a single case of Mouton-Rothschild, and a variety of good-but-not-Harlan American wines - cabs, pinots, etc. He’s only 4 so it’s hard to predict if they’ll be for him or for me
I like the mixed-case approach, personally. I’d rather him have something that he likes, instead of blowing it all on e.g. Chambertin just to find that he doesn’t care for it. More than a financial investment, I want the wine to be a meaningful connection - something that we can share and enjoy together.
I have bought about 300 birth year wines of many different grapes and regions that I like. I would say the average per bottle cost comes at $200.
Adopt me?
I have a small buttress of wine in mags stacked on the floor of my cellar consisting of 1.5 L bottles for my 3 children’s birthyears. I had to be flexible and pick and choose based on what seemed to do well in the particular vintages they were born during. Champagne, Bordeaux and Ca Cabs mostly----but it’s amazing what you can find if you ask. For example a Super Tuscan I really enjoy year after year is Ilatraia from Brancaia. Had no idea they even make mags of it but an email to the winery confirmed yes they do. When the first shipment was somehow lost in the mail they replaced the 2 I had purchased and added a third for free as a compliment for “my troubles”—their interpretation not mine as I intended to put the bottles away for another 14 years so a couple of months waiting on a lost shipment didn’t cause me any stress. But regardless great customer service and yes again—mags mags mags!
What are these 1- and 2-liter bottles you keep talking about? Mags I understand, but virtually all 1-liter bottles are some cheapies and can’t remember ever seeing a 2-liter bottle.
Radikon does 2L bottles. Maybe his kids will be into it?
For my son we opened birth year wine on his 21st. He still talks about how special that was. A 97 Seavey and a Crystal that was a bit older but I think i picked up as a current release in 1997.
I have several Ports as I know they will last a long time.
A few scattered single bottles.
I bought also a Double Magnum of Les Pavots to open when he gets married. Several more magnums and had the wine makers sign them for him. Probably pop one of the magnums when he graduates med school.
He looks forward to when we open the scattered bottles and people enjoy the stories.
My daughter only will drink champagne so have some 2002’s for her. Glad i didn’t buy a bunch of big red’s.
I got these for my kids recently - one 3L bottle for each. The idea is to have a party and hopefully this size is about right. Pride was a highlight of a trip to Napa that was our last big trip as “DINKs” before our daughter was born. I love that the name/label can apply to how we feel about our kids. We’ve also stayed on their list / bought more than probably any other winery. The bottles are just beautiful with the etched labels too - kind of hard to see from my mediocre pictures…
For me, multiple bottles seem like a bit of a guess, or something more for me than for them, but large format enjoyed together is something I’ll be looking forward to for years. I do have a box of cigars for each (maybe actually need to buy the final box) which I smoke each year or not depending on whether I remember to, and hope at least the boys might join me when old enough.