The band's lead singer is called Elsbieta, a Polish version of Elizabeth which often has the nn Ela (the singer uses this nn). The band's web page says that the name should be pronounced Ela-i-sa. That suggests that the band's name is based on her nn.
Alternatively, Elaisa is the name of a house design in the Philippines. (Wonderful thing, Google.) So Elaisa -with-an-s might be a name created fairly recently in that region. In which case, as Hello said above, it might not have any meaning.
Alternatively again, Shahira Elaiza writes a blog about the arts and her Muslim faith: www.shaelaiza.com/ She might know more about her name.