Milo of Croton is said to have attained enormous strength by, as a child, lifting and and carrying a newborn calf.
This feat was repeated daily and as the calf grew, so too did his strength.
Newborn calves are generally lacking these days but there are plenty of newborn humans available, especially to expecting couples.
Babyweights would consist of a set of ripstop nylon sacks. That's pretty much it. The sack would have handles placed on it in various configurations and possibly holes in it for arms, legs and a head.
Then, you place a baby of your choosing in the sack and work out with it. Fresh babies are rather light so single-limb isolation exercises done with high reps would be initially appropriate.
As the baby ages (and grows heavier) you can start using some of the alternate handle placements on the bag and switch to compound movements -- things you would have been accomplishing with a barbell. Worn as a backpack, you could do lunges, squats and good mornings to hit the legs and back.
Babyweights would be sold in sets by age - 0-1, 1-3, 3-5 and so on.
When the baby reaches the age of fifteen or so, you can switch to the Babyweight Sr., which would basically be a body bag with extra handles.
By the time your baby is in their twenties, you could potentially be looking at 200+ pounds of weight. This is excellent for heavy squats and building strength via low-rep work.
Thanks, Milo of Croton!