From Alfred's (10) marriage and census entries it appears fairly certain that his mother was Elizabeth.
On Alfred's birth certificate his mother is given as Elizabeth Cox, formerly Brookes.
There is indeed a marriage for Samuel Cox and Elizabeth Brookes in 1830.
Likewise there is a baptism for Elizabeth consistent with the age given in 1851.
Therefore the later identify of Elizabeth seems fairly certain.
However, the early identification of Elizabeth and her parents is cicumstantial and nees some corroboration.

Her ages at the Censuses are inconsistent meaning she was either born about 1808 (C1851) or 1804 (C1861) in either Aston or Birmingham.
There are two possible baptisms:
Baptism St Philip, Bham 
 - 18 Apr 1808 to Thomas and Sarah Brookes.
Baptism St Philip, Bham  - "Eliza" to William and Martha Brookes. born Sep 15 1807
The former looks most likely but needs corroboration.


  

The 1871 Census has been examined but no obvious entries for Elizabeth Cox found.
Searching for entries in the Death index for Elizabeth revealed several possible entries that need to be checked against the ages.