There isn't a way to just take damage barring GM fiat; below are what I found to be the least damaging methods depending on your situation:
If you want to use up your action
Your best option is probably a torch:
A torch burns for 1 hour, providing bright light in a 20-foot radius and dim light for an additional 20 feet. If you make a melee attack with a burning torch and hit, it deals 1 fire damage.
This has you only taking 1 single point of damage to proc the invisibility.
Alternatively as user @NameDisplay mentioned in a comment, you could throw caltrops onto the ground:
As an action, you can spread a bag of caltrops to cover a square area that is 5 feet on a side. Any creature that enters the area must succeed on a DC 15 Dexterity saving throw or stop moving this turn and take 1 piercing damage. Taking this damage reduces the creature's walking speed by 10 feet until the creature regains at least 1 hit point.
A benefit of this option is that it works even if you are resistant to fire damage and a Thief Rogue could use these as a bonus action. It does require at least 5 movement though which is unfortunate.
If you want to use up your bonus action
Your best option besides a Thief Rogue, is probably an attack with either an unarmed strike or a blowgun somehow. One way is a Monk using Martial Arts. The first option lets you deal 1 + Strength modifier damage and the second deals 1 + Dexterity modifier damage.
If you want to use up your movement
Your best options (besides fall damage, and walking into AoEs) is... well, you don't really have one. The only method I can think of is provoking an opportunity attack, but this could deal a significant amount of damage.
One thing that might be possible is to take fall damage from jumping, if your GM rules that you can do this, then you would be limited by the high jump rules:
When you make a high jump, you leap into the air a number of feet equal to 3 + your Strength modifier (minimum of 0 feet) if you move at least 10 feet on foot immediately before the jump.
You would thus need some way of increasing your jump height to be at least ten feet (the minimum to take fall damage). You could be a level 20 Barbarian (Strength modifier of 7), a Monk (Step of the Wind), or simply be under the effects of the jump spell. Unfortunately, the latter two require your bonus action even if you have a Ring of Jumping.
However, there is a magic item that increases your jump height without requiring anything but attunement: the Boots of Striding and Springing:
While you wear these boots, your walking speed becomes 30 feet, unless your walking speed is higher, and your speed isn't reduced if you are encumbered or wearing heavy armor. In addition, you can jump three times the normal distance, though you can't jump farther than your remaining movement would allow.
This would allow any character with a Strength modifier of at least 1 to jump 10 feet into the air and thus take fall damage. This would however, also require a running start; if you had a strength modifier of 4+ you would not need to make a running start.
If you want to use up your object interaction
You don't have any options. The PHB lists some possible things to do with your object interaction, and none of them come close to dealing damage:
- draw or sheathe a sword
- open or close a door
- withdraw a potion from your backpack
- pick up a dropped axe
- take a bauble from a table
- remove a ring from your finger
- stuff some food into your mouth
- plant a banner in the ground
- fish a few coins from your belt pouch
- drink all the ale in a flagon
- throw a lever or a switch
- pull a torch from a sconce
- take a book from a shelf you can reach
- extinguish a small flame
- don a mask
- pull the hood of your cloak up and over your head
- put your ear to a door
- kick a small stone
- turn a key in a lock
- tap the floor with a 10-foot pole
- hand an item to another character
But wait! There may be a way to do this
User @goodguy5 jokingly mentioned taking damage from drinking a flagon of ale, and it may in fact be possible to do so (I have now asked whether this is possible here)!
There are many poisons and one is the Assassin's Blood (Ingested) poison which states:
A creature subjected to this poison must make a DC 10 Constitution saving throw. On a failed save, it takes 6 (1d12) poison damage and is poisoned for 24 hours. On a successful save, the creature takes half damage and isn't poisoned.
Though this does have a significant negative side-effect should you fail the saving throw (poisoned for 24 hours), it does do a surprisingly small amount of damage and the DC is quite small. If you were proficient in Constitution saving throws and had a high Constitution modifier, you could near-guarantee a success.
The section on ingested poisons state:
A creature must swallow an entire dose of ingested poison to suffer its effects. The dose can be delivered in food or a liquid. You may decide that a partial dose has a reduced effect, such as allowing advantage on the saving throw or dealing only half damage on a failed save.
Thus, depending on what your GM says, you could theoretically poison a drink and then drink it using your object interaction, resulting in you taking damage. Unfortunately this would require two object interactions: one to get the drink out from wherever it is, and one to actually drink it.
GM dependent ideas
There are other possible things your GM might let you do with an object interaction such as slamming you hand in a door or extinguishing a flame with your hand. Ask your GM about these sorts of things.