Based on Thomas' Social Security record above, his mother's last name was Gibbons. Bridget was born in New York. I found a Bridget Gibbons living in New York in 1880. She might be your Bridget Mellick? Her father had passed away. Both parents came from Ireland. Did not find where they came from.
1930 Census
NameBridget Mellick
Birth Yearabt 1862
GenderFemale
RaceWhite
Age in 193068
BirthplaceNew York
Marital StatusMarried
Relation to Head of HouseWife
Homemaker?Yes
Home in 1930Newark, Essex, New Jersey, USA
Map of HomeNewark, Essex, New Jersey
Street AddressClark St
Ward of City1
Block53
House Number9
Dwelling Number2
Family Number5
Attended SchoolNo
Able to Read and WriteYes
Father's BirthplaceIreland
Mother's BirthplaceIreland
Able to Speak EnglishYes
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1880 census
NameBridget Gibbons
Age19
Birth DateAbt 1861
BirthplaceNew York
Home in 1880New York City, New York, New York, USA
StreetMulberry St
House Number120
Dwelling Number17
RaceWhite
GenderFemale
Relation to Head of HouseDaughter
Marital StatusSingle
Father's BirthplaceIreland
Mother's NameBarbara Gibbons
Mother's BirthplaceIreland
OccupationTobacco Worker
NeighborsView others on page
Household Members (Name)AgeRelationship
Barbara Gibbons37Self (Head)
James Gibbons15Son
Thomas Gibbons10Son
Bridget Gibbons19Daughter
Regards- Mary