I have an idea for a weapon that can permanently absorb items that require attunement to gain its properties (including its attunement requirement), up to 3.

For example, the item alone does not require attunement.

It then absorbs a pair of Boots of Speed. We will ignore the requirement of "click your boots together", and instead the user must take a bonus action to perform an appropriate task (perhaps tapping the weapon on your legs) to activate the item's Boots of Speed property, which gives the normal benefits of the boots of speed. The item now requires one attunement "slot" (of 3).

Further attunement items absorbed into the initial item gain their properties and raise the attunement requirement, up to 3.

Because artifacts are of nebulous power, and therefore have not many guidelines, the following question pertains to only the rarities common though legendary.

Is this property inherently imbalanced? If not, what would the rarity be of an otherwise mundane weapon with this property?

Since this item does not let you evade or increase your Attunement Slot limit, the main way in which it increases your power is that it would allow you to wear multiple items that go in the same body "slot". The Magic Items section of the Players Handbook says:

Multiple Items of the Same Kind

Use Common sense to determine whether more than one of a given kind of magic item can be worn. A character can’t normally wear more than one pair of footwear, one pair of gloves or gauntlets, one pair of bracers, one suit of armor, one item of headwear, and one cloak. You can make exceptions; a character might be able to wear a circlet under a helmet, for example, or to layer two cloaks.

With this item, it would be possible, in theory, to gain the benefits of multiple boots or hats or cloaks at the same time. It's hard to say how powerful this is, as it depends on the specific magic items to be combined. I am sure that there are extremely powerful combos that would be enabled by this, but I don't think it's inherently unbalanced.

You might want to prevent this weapon from absorbing powers from other weapons. More than other items, weapon abilities are balanced around the idea that you'd only benefit from 1 (or 2) at once. Combining a Dwarven Thrower, a Flame Tongue Sword, and a Holy Avenger, for example, would give you a +6 weapon that does +1d8 +2d6 +2d10 (vs undead) bonus damage on each hit.

You might also want to address whether identical bonuses stack. For example, if you combined 3 copies of the Cloak Of Protection, you would add +3 to AC and all saves. I wouldn't say that's overpowered (since it uses all 3 of your attunement slots) but you could imagine that combining more powerful items could generate very large bonuses that break bounded accuracy.

Yes, for a few duplication reasons

First, unless a restriction was added, you would be able to attune to same item multiple times. Picture attuning three Staff of the Magi, with all the bonuses, charges, and other factors tripled.

Second, you would be able to attune multiple items that would normally take up the same "slot" on the character. Per the DMG:

Multiple Items of the Same Kind

Use common sense to determine whether more than one of a given kind of magic item can be worn. A character can’t normally wear more than one pair of footwear, one pair of gloves or gauntlets, one pair of bracers, one suit of armor, one item of headwear, and one cloak. You can make exceptions; a character might be able to wear a circlet under a helmet, for example, or be able to layer two cloaks.

So the player could have multiple cloaks active at the same time. Imagine walking around with the powers of the Cloak of Invisibility, Elvenkind, and Protection at once...

Finally, weight may play a big factor. The one object would have a set weight of X. But the three attuned items may have a combined weight that even a giant would struggle with.

The big question we need to ask to work this out is the following:

"Is there a combination of Magic Items that interact differently with each other in the strange scenario where they are, themselves, the same item? And if so, is that situation gamebreaking/unbalanced?"

I can think of a few.

Weighed Down

My first instinct is to point out the fact that this item elides any excess weight the combined items would otherwise have. If this weapon gains the weight of every item it absorbs, that will probably make it unsuitable for actual use as a weapon, so we have to deduce that this doesn't happen: it gains the abilities and features of the items it absorbs without gaining [excess, undue] weight. That could represent a significant buff in power for certain kinds of heavy Magic Items, especially magical armors.

Speaking of which...

The Sword of Many Swords

A common limitation on magical items is the fact that two items that are of the same type (two sets of Body Armor; two Helmets; two sets of Arm Bracers) cannot be used at the same time. Even if there's no implicit restriction on their ability to stack, the fact that Humanoids in 5th Edition D&D cannot wear more than one set of armor at once means that this item could absorb, for example, multiple kinds of Body Armor and confer the effects of all of them simultaneously.

Alternatively, it could absorb three different kinds of weapons, and take on all the properties of each of those weapons simultaneously. In addition to needing to adjudicate how those properties interact (if all three weapons have +Attack/+Damage buffs, do all those buffs stack?), it could also get very powerful very quickly: Combine a Scimitar of Speed ("You may make a single Melee Weapon Attack with this weapon as a Bonus Action"; DMG pg. 199) with something like a Frost Brand weapon or a Flame Tongue weapon and suddenly you get a free 3d6 damage on a weapon that also can make extra attacks each turn with a Bonus Action. That wouldn't be possible for a character that simply had each of those three weapons; they'd have to switch between them between attacks (only one Free Item Interaction per turn!) and even then they'd only get the benefits of each weapon on the specific attack that used that weapon.

Overpowered?

Probably.

For all the reasons I've listed above, if I were to put such an item in my games, it would definitely be at least Legendary quality, maybe Artifact tier. I mean, this is an item that can absorb other Magic Items. That's some supremely powerful magic, even if this itself excludes other Artifact-tier items.

There's definitely some powerful ways to exploit the power of an item like this, which I've only sampled here. So what it really comes down to is the question of whether or not that kind of power is acceptable in your campaigns. That might very well be the case, especially since as DM, you do have control over what kinds of Magic Items get distributed in the first place. If you're in a campaign where A) players get enough Magic Items to make an item like this worth having in the first place, and B) they've managed to acquire such a powerful artifact-like object anyways, then chances are pretty good that you're already in a high-magic setting where this doesn't really change much.

There is the possibility that there exists some combination of items that normally couldn't be used at the same time. Something like the two following items:

Helmet of the Petrifier

Wondrous Item, Legendary (Requires Attunement)

As an Action, you may gaze at a target within 60 feet. That target must succeed on a DC18 Wisdom Saving Throw or be Petrified until the beginning of your next turn.

Helmet of the Shatterer

Wondrous Item, Legendary (Requires Attunement)

As a Bonus Action, you may gaze at a solid object no bigger than a 10' cube, or a petrified creature, within 60 feet. That target is immediately destroyed and reduced to rubble.

—Invented on the Spot Magic Items to Prove a Point

Neither of these items are absurdly powerful on their own (at least for Legendary Items) but combined, they're exponentially more powerful. Under the normal Magic Item rules, it would be impossible for a normal creature to wear both of these at the same time, but with your item, it would certainly be possible to use both of their effects at once.

So in general, this is a very powerful item; but like I said, if that's acceptable for your campaign, then go nuts.