Most wine at $10 or under is made to drink right away and won’t really age. They will just turn into vinegar.
The qualities that make a wine age well are acid and tannin. (Acidity in a wine refreshes you, like drinking cranberry juice or lemonade. Tannin gives you that mouth-drying feeling like when you drink very strong black tea.)
If a wine has lots of these, they will gradually soften over the years until you’re left with a wine that has complex, savory flavors and a silky texture.
On the other hand, most wines at $10 are made to be fun and fruity so that you can drink them right now. They generally don’t have that much acid or tannin, just ripe fruity flavors. That’s not a bad thing, but you’d probably be disappointed when you opened the time capsule.
If you really want to do this, and have something you can drink at the end, it is technically possible. Try a few Bordeaux wines at that price, and look for “structure,” the combined presence of nice amounts of acidity and tannin. If a wine tastes a little “harsh,” like you’d prefer to have it with a steak than on its own, then you might have a good candidate for aging. There are some hidden gems in Bordeaux with these qualities at this price. The Mouton Cadet Bordeaux that’s widely available in grocery stores is not one of them.
Wines from the South of France, like the Languedoc/Roussillon, Corbieres, St-Chinian, Minervois, etc., also often have these qualities at this price.
This is only if you want to have something you can drink at the end of the time capsule. If it’s just a fun lark, then you can put any kind in there, but it will probably have turned by the time you pull it out.
If you go up to $20, you will be able to easily find a wine that you can age for seven years. Certain Bordeaux wines have cabernet sauvignon as their main ingredient, and they have more “structure” than cabs at the same price from other parts of the world. Look for the names “Medoc” or “Haut-Medoc” especially (this is a big subset of Bordeaux, not an individual producer. I don’t know availability in your area, and it’s easier to find types of wine than individual wines.)
2009–2010 should be the current vintages available for wines like this, and both are awesome and would be a real treat to open in a few years. Store the wines in a cool dark place and on their sides (very important!), so the cork doesn’t dry out over time.