die Eigenheit
quirk
A special, often strange or unusual trait or habit of a person or thing.
Es ist eine Eigenheit von ihm, beim Nachdenken immer mit dem Stift zu klicken.
It's a quirk of his to always click his pen while thinking.

peculiarity
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quirk
A special, often strange or unusual trait or habit of a person or thing.
Es ist eine Eigenheit von ihm, beim Nachdenken immer mit dem Stift zu klicken.
It's a quirk of his to always click his pen while thinking.

The word is derived from the adjective 'eigen' (own) and made into a noun with the ending '-heit' (-ness). So it describes a characteristic that is 'eigen' (specific/particular) to someone or something, meaning that specifically belongs to it.
Die Eigenheit dieses Materials ist seine extreme Hitzebeständigkeit.
The peculiarity of this material is its extreme heat resistance.
When 'Eigenheit' (peculiarity/quirk) refers to a person, it often means a strange habit. When it refers to a thing, a language, or a region, it describes a neutral, characteristic feature.
Seine Eigenheit, Sätze nie zu beenden, ist manchmal nervig.
His quirk of never finishing sentences is sometimes annoying.
When 'Eigenheit' (quirk) describes a habit, it is often expressed with the structure 'Es ist eine Eigenheit von jemandem, etwas zu tun' (It is a quirk of someone's to do something).
Es ist eine Eigenheit von ihr, vor wichtigen Terminen immer ihr Büro aufzuräumen.
It's a quirk of hers to always tidy her office before important appointments.
In the meaning of 'characteristic feature', the word is often used in the plural and with adjectives such as 'sprachlich' (linguistic), 'kulturell' (cultural), or 'regional' (regional).
Der Reiseführer erklärt die kulturellen Eigenheiten des Landes.
The travel guide explains the cultural peculiarities of the country.
characteristic / capacity
'Eigenschaft' and 'Eigenheit' both describe something that is typical of a person, thing, or place. 'Eigenschaft' is neutral and general. It can refer to good, bad, or technical characteristics. 'Eigenheit' sounds more particular and often a bit more personal or unusual. It can be a small strange habit or a peculiarity that distinguishes something from other things. The words can be confused because both talk about characteristics. For neutral descriptions, 'Eigenschaft' is usually the right choice. When something seems noticeable, special, or a bit peculiar, 'Eigenheit' is more precise.

Eine Eigenheit unseres Nachbarn ist, dass er jeden Abend seine Pflanzen zählt.
One quirk of our neighbor is that he counts his plants every evening.
Geduld ist eine wichtige Eigenschaft, wenn man mit Kindern arbeitet.
Patience is an important characteristic when you work with children.