This may be a little behind the rest of the discussion but I thought I would bring it up anyway. I haven't really researched Lagrange points etc. but I do know a little bit about this subject (because teaching requirements for life around a star was part of my curriculum as a teacher in Texas a number of years ago).
An important aspect of having a planet that is tidally locked is the location of the "life zone" around the star. A star's class and color are directly correlated to star temperature. The colder the star, the more red.
Basically K (orange) and M (red) class stars would be ideal for your setting. These (especially M) are the coldest stars and because of that, the "life" zone is fairly close to the star. Keep in mind that these stars are also smaller than our own.
Another reason these stars are a great setting for tidal locking is because tidal locking requires a lot of time to occur. (Mercury may eventually become fully tidally locked). Red dwarfs (M class stars) last 20 billion years or more before they "die." So there is plenty of time for tidal locking to occur. K class stars last around 15 billion years. One thing to note though is that tidal locking would require more time around a K or M star than a G star with a given distance because the star has less mass.
EDIT: One thing to note about M and lower mass K stars is that they tend to be metal poor. This might be important to your story.
thank you for this bit of information, have a bit of thinking to do now as you've made an excellent point about the different types of stars.