Yes, big(1)/100 is correct and no need for strings, but if one wants consistent zero-padding for a random Float64 then the conversion to string and then BigFloat seems to work consistently:
julia> a = rand()
0.6978961477555439
julia> BigFloat(a)
0.69789614775554387193778893561102449893951416015625
julia> BigFloat(string(a))
0.6978961477555438999999999999999999999999999999999999999999999999999999999999987