Thank you for your response. I took another look at the 1860 census. The Rebecca mentioned in the household had to be the wife of Benjamin. I missed a very important part of the census when I first looked. The column "married within the year" has a mark beside Benjamin and Rebecca. I've not been able to locate them since the 1860 census. I don't think that is the Rebecca in the household with Laura as her grandmother.
I'm sure the Beck you mentioned is Bican/Bical/Bikile P. Cunningham. She is listed as mother-in-law in the household of John Thomas SMITH and his wife Delilah on the 1900 Lamar Co., AL, census. She is also listed in their household in the 1920 census Fayette Co., AL, census. The oldest daughter of John Thomas Smith and wife Delilah/Lillia on the 1900 census was Laura F. I had been unable to find Bican in the 1910 census and certainly appreciate your information. If you ever figure out where the name Bican/Bical came from, please let me know. That is one of the strangest names I've run across.
Another tidbit for you. According to the death certificate of Mrs. B. P. Cunningham, her parents were Jesse Cunningham and Delilah Jones. Note the fact that her death certificate has her listed as "Mrs.". It is my understanding that she was never married. She is listed in the 1850 Shelby Co., AL, census in the household of Jesse and Delilah Cunningham. So far, I've not been able to find further proof of her parents names. My past experience with death certificates has taught me that the information often has errors.
This discussion could prove lengthly. If you are interested in exchanging further information, please e-mail me.