Does "peculiar" imply "unusual"/"hard to expect"?

Solution 1:

Pecularity means "an odd or unusual feature or habit", but also "a characteristic or quality that is distinctive of a particular person or place".

For all her peculiarities, he finds her personality quite endearing.
Her essays characterized decency as a British peculiarity.

Peculiar means "strange or odd", but also "unusual", or (in peculiar to) "belonging exclusively to".

Her accent was a peculiar mixture of Cockney and Irish.
The air hung with an antiseptic aroma peculiar to hospitals.

Peculiar and having a peculiarity have both the meanings you supposed.

Informally, peculiar means also unwell: she felt a little peculiar for a while.

_{[Reference: the New Oxford American English.]}