Since JLC's first appearance in "Asylum of the Daleks," I've noticed that the colour red has been a part of each version of the character. Whether there's anything deeper than just a cursory bit of continuity, I'm not sure, but it's kinda cool nonetheless. Oswin Oswald: sexy red dress, red sneakers, red stripes on socks Clara Oswin Oswald: first half of episode she wears a deep red dress Clara Oswald: the most important leaf in the universe is red; she's wearing a red jacket when the Doc meets her as a kid; at her mum's funeral, she's wearing a red coat; when we first meet present day Clara, she's wearing a red/black top; she wears red leather gloves on the motorbike, her red satchel is a continuing accessory. Is there something there or do I just have too much time on my hands? Maybe she just likes the colour red!
I think it's a very deliberate choice by the costume department to give Clara her own visual signature, but I don't think there's any narrative reason for that choice.
Of interest is that Clara has died (twice!) and doesn't know it; this puts me in mind of The Sixth Sense which also marked dead characters with the color red and used it to foreshadow the final twist. Perhaps it's a reference, or perhaps it is significant some way.
Heh. Well, I think it's more significant that red was a hint that Bruce Willis was dead all along and didn't know it – which is a situation much akin to the one that Clara finds herself in.
And Amy had something red in most episodes, especially in her first season. It may just be a choice of the actress giving the character a favorite color rather than some deeper meaning.
I think they're just giving the character a color theme. That's not unusual to begin with but they may particularly want one for Clara in order to draw the connection between her separate selves. I wouldn't read too much into it though. Mr Awe