Say I start off with N single cells in a sample. These are then allowed to grow untill there are x progeny from each individual cell. How many cells must I take so that I have at least one progeny from each the N original cells in the sample?
You have a total of $Nx$ progeny.
You could select $(N-1)x$ without having at least one progeny of every original member by having all progeny except those of one particular original cell.
So to be sure to get at least one of each, you need to select $(N-1)x+1$. Anything less can leave you missing one type.
As for the relationship to the relevant sock drawer problem, the problem works like this:
In your sock drawer, you have 2N socks from N distinct pairs, all as single socks. The light in your room isn't working, and you need to grab a pair of socks, so you plan to take some socks out to the light to find a pair from. What's the smallest number of socks you need to take to be sure of finding at least one pair?
(This is similar to your problem - you need to cover the case where you get one sock from every pair - and then take one additional sock, which would then match one of the already selected socks. That is, you need to take N socks in case you get get one from each pair, and then one more sock will be certain to form a pair. Any fewer could leave you with all single socks. The count of "once sock from every pair" is where it's very much like yours, but the form of problem is slightly different)