It's an invented word. It could relate to the bird or, if the text relates to the UK/former British Empire, it could possibly mean a dress (like or shaped like) those worn by the "Wrens" = members of the UK's former Women's Royal Naval Service (WRNS) – although their uniforms were blue.

Wren
1) A girl, usually with brown hair, that has a happy-go-lucky look on life. She's pretty book smart, but sure knows how to party. With a good smile, she is the type that makes her best guy friend fall in love with her. And every guy that she knows, will at least once have a thing for her.

5) Skinhead girl, variation on the term "bird."
The skinhead talked only with his wren paying attention to no other girls. By her short hair and boots she was identified as a wren.

9) Wren is a person who has very low self esteem, but never lets that show. Wren uses humor to mask their insecurities. Wren is an hilarious, smart, and outgoing person who loves to make someone laugh. Wren has a lot of problems with love life, but doesn't worry too much about it. Wren is someone who is a shoulder to cry on, no matter how broken they are.

Here's a sentence written by a native speaker of English (Ronald Wright) that contains the term 'wrennish':

The cathedral's design is oddly Wrennish — Ionic columns along the facade and a blue tiled dome.

Hmmm.... Your No 1 may be right, although I'd always beware of the Urban Dictionary as there's so much rubbish in it. However, the passage actually describes her dress, rather than her, as "wrennish", so it could just mean 'the colour of a wren [bird]' as others have suggested.

If you want to explore the context further, you can see more of the book here: http://books.google.co.uk/books?id=YpvNBAAAQBAJ&pg=PT20&lpg=PT20&dq="her+dark+brown+eyes+set+the+tone+for+her+wrennish+dress"&source=bl&ots=hZDiZnNPkw&sig=25J12cmmSVXsN2EeIcqseYu2jvI&hl=en&sa=X&ved=2ahUKEwjszIbmt-jcAhWLmLQKHfXMDIIQ6AEwAHoECAAQAQ#v=onepage&q="her dark brown eyes set the tone for her wrennish dress"&f=false

How long does it take to type 'Спасибо'/'Спасибо всем'?