Ah'Kiell Walker died at Bristol Children's Hospital on Sunday.
His parents Hannah Henry, 20, and Alistair Walker, 26, of Archdeacon Street, Gloucester, have been remanded in custody accused of child neglect.
In a statement, Ah'Kiell's family said: "Our days are so much darker without his smile, but we will never forget the times we had."
"We are blessed to have had our beautiful baby boy Ah'Kiell in our family, even if for such a very short time," they added.
"He was taken too soon but his beautiful eyes, smile and amazing hair will always be in our hearts.
A police investigation into Ah'Kiell's death is continuing.
The family of a three-month-old old boy who died at the weekend have said he was a "beautiful baby".