The image posted is from a transmitted record. I do think our OP should follow the advice given by JenB and seek out the original register.
There's several possibilities for that July 1814 baptism:
a) Elizabeth is an adult, and the clergy has only asked for her father's name as a way of registering her details in the parish register, anticipating that few people would need to consult the register
b) Elizabeth is a child aged seven or more (I stand corrected on 'age of seven' relying on grey cells), who has gone to the clergyman directly and asked to be baptised and the clergy have agreed that she understands this sacrament (one of only two sacraments in the C of E) and the clergy has only asked for her father's name as a way of registering her details in the parish register (as few needed to consult the register)
c) the scribe transferring the information into the Bishop's transcripts has not been able to decipher the name of the mother from the transmitted record sent through.
d) the scribe has not re-check, nor has the scribe had anyone re-check the transcripts sent through
e) the transcription sent through from the original register did not record mum's name due to an oversight
f) the transcription sent through from the original register did not record mum's name because mum's name was not known to the clergy
g) and h) and probably many others as well ......
So please consider getting to the original register
Durham Records Online has the South Shields St Hilda baptisms for this year, and according to their Parish Records page http://www.durhamrecordsonline.com/Parish_Record_Info.php their record is taken from the original rather than the BT's.
You could buy a few credits to see if a mother's name is mentioned on there - but a search of the index cross-referenced with the mother's name Cecilia gives a negative result.
If there's no mother's name on the original I'd tend to go along with what others have said assume that the omission was an administrative error.
PS ADD
we need to remember that it was not until the 1960s or there abouts that baptisms and/or civil registrations commenced to record a surname for the person being 'registered'.... It is logical to assume that Elizabeth's surname was WILLIAMS, but the baptismal record does NOT actually NOTE that, particularly as her mum's name is not on it. Many of us overlook that a baby's surname comes from their mum's then surname, particularly in the 19th Century .... If mum is a married woman, then likely she would be using her husband's surname as her own.
ADD
....... This taken from 'durhamrecordsonline' but there is no Mothers name.
Cheers, JM (edit to sort out some grammar)
