The name Jessamine is a girl's name of Persian origin. Jessamine and is often added to lists like Flower Names for Girls and discussed in our forums with posts like "A brother or sister for little Jack or Olivia".
From the experts:
Jessamine, a charming name occasionally heard in England, is just beginning to be appreciated in the U.S. as a possible successor to all the Jess names of the past. It's also spelled Jessamyn, as in Quaker novelist Jessamyn West, author of Friendly Persuasion--who started life with Jessamyn as her middle name.
Jessamine began to be used as a first name in Britain around 1900, along with other flower-related names. Yasmin and Yasmine are other Jasmine variations.
This would make a great choice as a namesake honoring a Jessica.