In Dickens' Oliver Twist, what clue remains of Oliver's parents?

Oliver’s mother had a locket and a wedding ring, which Sally stole and Mr. Bumble’s wife took. 
Oliver's mother barely made it to the workhouse before she died shortly after giving birth to him.  Oliver was the product of her affair with a man named Leeford.  Sally stole the locket from her, but on her deathbed she felt bad about it and confessed.  That is how the Bumbles ended up with it.
Bumble meets Monks (Leeford's son) and takes him to show him the locket, which contains locks of hair and a ring.  It also has Oliver’s mother’s name, Agnes, written on it.  Although this was before the days of genetic testing, the locket has enough information that it could be used to prove Oliver’s parentage. 

… It contained a little gold locket: in which were two locks of hair, and a plain gold wedding-ring.
'It has the word "Agnes" engraved on the inside,' said the woman.
'There is a blank left for the surname; and then follows the date; which is within a year before the child was born. I found out that.' (Ch. 38) 

Monks knows that the locket and ring will prove that Oliver is his half-brother, so he throws them in the river.  Monks's efforts are all in vain, though.  Nancy tells Rose Maylie that Monks hired Fagin to turn Oliver into a thief.  
Brownlow tracks down Monks, who explains that he knew about his father's will and tried to make sure that Oliver never came to light.

He reminded her of the day he had given her the little locket and the ring with her christian name engraved upon it, and a blank left for that which he hoped one day to have bestowed upon her... (Ch. 51)

Oliver was able to remain with Brownlow.  He got to have the lifestyle he deserved and have a family.  Monks tried to keep that from him, but was unsuccessful.  One of the reasons he was unsuccessful is that Oliver was good and remained good, no matter what bad influences he had.