I know this is an old post, but I was doing a search and found this thread. I'm stuck on these two too. They are my great great grandparents. I've been talking to my uncle, their grandson, who was told by his mother that Rev John Green found Frank crossing the Red River as a young child and spoke no English - I presume he took him into his care since family members have said he was a worker on the farm. He also told me the possibility of the name originating from the French/Spanish/Seminole indians from FL and the name was originally spelled Chavedoux (or maybe Chavedeaux).
The ONLY record I could find is their marriage - Francisco Chavida & Rebecca A Green, 13 Nov 1883, Uvalde County, TX. That's all the information it gives - I'm thinking of going down to Uvalde and searching the archives since they don't do research for you. I've tried spelling the name every which way possible and can't find anything nor can I find a record for Beckie's birth. Have you found anything new?