So, players want to have some means of healing more than 1 or 2 hit points per day, when they're 1st level and have no healing spells. Some Referees solve this by making potions of healing easily purchased, and in fact they're listed right on the equipment list in the 5th Edition Player's Handbook. While I get the desire for some help in a low-powered game like mine, having this in 5e - where even fighters can heal themselves, using some pseudo-feat called "Second Wind" - is a bit excessive. (This also points to my oft-repeated lament that there is no such thing as a plain fighter in 5e, but that's beside the point.)
My old reliable game, B/X, had an interesting idea: giant bees make honey that is so nourishing, it serves as a weaker form of a healing potion! One's head swarms (heh) with the possibilities of the players going off to raid a huge nest of giant bees; if they succeed, they'll be better equipped for future adventures, at the risk of dying horribly from toxic shock and full-body edema. In fact, in my setting, it's now official that potions of healing use giant bee honey as the active ingredient.
But I wanted to address the desire for some extra healing, for those situations where luck is emphatically on the monsters' side. I found a bit of inspiration in the game I've been playing recently, Fallout: New Vegas. I play it on Hardcore Mode, where healing items such as Stimpacks and food don't instantly restore health, but do so over a span of several seconds. In a game with real-time combat, watching both damage and healing ticking up at similar rates creates a lot of tension, something that is hard to duplicate in a tabletop game. The idea of gradual healing stuck with me, though, and the final inspiration came from the bizarre first-aid item in The Ring: Terror's Realm known only as "Healing Jelly". (Spoony's mockery of this is what made it jump out to me.)
Item: Healing Jelly
Cost: 50 gp
Weight: 1 lb
This substance, made from the honey and wax produced by giant bees, is sold in small glass jars to prevent it from drying out. Whenever a person eats the jelly or rubs it on their wounds, the rate at which their body naturally heals itself is doubled. (In Basic Fantasy RPG, 1 hit point is restored per day of normal sleep, and 1 additional hit point is restored if they have full bed-rest; so a day of full bed-rest after eating and/or applying healing jelly would restore 4 hit points.) A jar contains enough healing jelly to treat one Small- or Medium-sized creature, or two Tiny-sized creatures, for one day.
Healing jelly is not considered a magical item, and is not sensed by Detect Magic spells, though it may be located normally with spells such as Locate Objects, or a crystal ball.