"Her Bright Smile Haunts Me Still" seems to be quite a popular song still--lots of clips on YouTube, including several of a setting quite different from Wrighton's. My favorites:
Walter "Salty Walt" Askew (solo unaccompanied)
Hughes Macklin and David Brazell (from an Edison cylinder)
As well as:
Mudcat thread: Her Bright Smile Haunts Me Still
DigiTrad entry: Her Bright Smile Haunts Me Still
Sheet music in the Lester S. Levy Collection

The Levy collection also has these songs set by Wrighton:
Sing Me and English Song
Oh! Chide Not My Heart
Dearest Spot of Earth to Me Is Home
Bright Star of Eve, Arise

Library of Congress American Memory collection has:
She Sang Among the Flowers (Wrighton; piano arr. by Hartl)
You Need Na Come Courting Me (Wrighton; piano arr. by Hartl)
Her Bright Smile...
Dearest Spot...
The Heart Should Have Some Tuneful Strings

Indiana University has
Her Bright Smile...
Norah, Sweet Norah

And they give his full name and dates: William Thomas Wrighton (1816-1880)