I am working on the premise that the child Samuel is with his uncle Samuel in 1881 as I mentioned- as Griffin.
1891 he is Samuel Rodgers with his mother and William Rodgers.
You could look for a marriage to check maiden name or get a birth certificate.
A possible marriage could be Dudley March 1850 vol 18 og 367
Samuel Griffiths and Elizabeth Sickard or Elizabeth Williams are on the same page - there are other names though but no more Elizabeths.