
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.

peculiarity
A characteristic feature or special quality that distinguishes something from other things.
Die sprachlichen Eigenheiten dieser Region sind sehr interessant.
The linguistic peculiarities of this region are very interesting.
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.
special feature
Besonderheit refers to a characteristic that distinguishes someone or something from others, usually without a negative connotation. Eigenheit more often means a typically individual trait or habit and can sound slightly odd. Besonderheit is appropriate for a neutral characteristic, while Eigenheit is more suitable for an unusual personal trait.

Eine Eigenheit von Paul ist, dass er seine Stifte immer nach Farben ordnet.
One of Paul's quirks is that he always sorts his pens by color.
Eine Besonderheit von Paul ist sein außergewöhnlich gutes Gedächtnis.
One of Paul's special features is his exceptionally good memory.