Materials elastic size

I don't know if there's any standard for string size per bjd size?
As long as the string is round and strong enough to hold the doll together it can work. The size is not usually super important but the doll joint style affects what works. I think I got maybe 4mm for 60cm SD and around 3mm thick for MSD but depending on the size of the holes I just go as big as possible. If the doll has a tendency to flop around it feels to me like thicker string can help make the doll stiffer.
I've noticed for example for a knee, if the string is (while tensioned) very close to not fitting in the string slot in the back of the knee when trying to bend the knee, the string itself will act as an extra stopper in the joint, if that makes sense? So you have to kind of make an effort to move the joint past that point. It helps to keep from bending a knee accidentally and I personally like that, but others might not like it and would prefer thinner string! It's not an exact science
Depending on what materials you use it's likely safest to start with thinner / weaker string though so you don't break anything while testing joints. The string tension can be a lot with a finished resin doll but paper mache etc might not take the same string without cracking.
When trying to decide on the string holes / path size keep in mind the string is thinner when tensioned. You can use a smaller string with little tension for testing and in the final product a thicker string might be the best fit because it can be pulled more and ends up the same width when tensioned.
My vote goes for try things out and see what you like!

 

The main thing that you need to keep in mind when choosing elastic size for a doll you are making is where the pivot point is in the joint. A thinner elastic will move the pivot point closer to the inside of the joint while a thicker elastic will force it further back towards the outside of the joint.
As mentioned by sinlaean the elastic will be thinner when streatched, so in order to get the lighter tension you need to test string a doll made from a more fragile material (compared to the polyurethane of the finished product) you can use a thinner elastic as long as you remember to leave room in the joints and channels for the thicker elastic of the finished doll.

Test stringing with a very light tension is usually a good idea anyway. As a rule of thumb; a doll that pose well with light tension will pose even better with high tension and a doll that poses badly with light tension will pose even worse with higher tension. This is because any little problems with unbalanced joints will be amplified with higher tension, causing joints to snap into certain positions and having trouble keeping their inbetween poses nicely. Generally you'd want joints to hold any pose within their range equally well and that can only be done if the joint is well balanced around it's pivot point.
There are exceptions, of course, where workarounds such as intentionally unbalanced joints or even locks are neccesary to get the preformance that you want, but for gerenal purpous joints such as shoulders, elbows, knees, hands etc. you'd typically want a smooth moving joint, not a snappy one.

So to sum it up; the ideal thickness of elastic is the one that the doll is made to work best with.