I think the skipping a generation thing is a myth. But I do think you are more likely to have twins run in the family.
I have a friend who's a twin, and he has twin sons. (I already thought the skipping thing was a myth, but this kind of proved it. He is an identical twin, and so are his sons.)
The skipping a generation thing comes from recessive genes. Anything that's caused by recessive genes can skip a generation and can also not skip a generation. It can also skip multiple generations. This is because you need two copies of the gene to get the characteristic.
For example, ginger hair. My mum has ginger hair, I have brown hair and one of my daughters has ginger hair. You need two copies of the gene to have ginger hair, because it's recessive. I have one copy (if I didn't, I wouldn't be able to have a ginger child) but the other gene is dominant (in this case for brown hair) and determines my hair colour. Traits coded for by dominant genes show in each generation, but those that are coded for by recessive genes can skip generations - or not as the case may be. Someone who's ginger can have a ginger child, if the child gets two copies of the ginger gene. And if both parents are ginger all the kids are ginger. So it's not that it skips every single time, or only affects (say) every other generation, but traits skipping generations is a very real thing.
Back to the OP:
AFAIK identical twinning is random, but non-identical twins tends to run in families. It's the double ovulation that runs in families and is more common in some ethnic groups. Non-idential twins happen when two eggs are released instead of one. Sometimes this can just happen randomly but it can also run in families (i.e. genes that make people more likely to double ovulate). Identical twins come from the same egg, where the embryo splits in two at an early stage. Although the way this happens is known, the reason why it happens isn't fully understood and it appears to be random and not related to genetics. Anything that's random can happen more than once in the same family by chance. Also, genetics isn't the only thing that increases the likelihood of twin pregnancies. There are multiple other factors. And if double ovulation runs in a family, it doesn't happen every time they ovulate, so not every pregnancy in someone who's inherited this tendency will be a twin pregnancy. And sometimes women who haven't inherited this trait can double ovulate anyway.
Vanishing twin is where one twin dies at an early stage of pregnancy and is reabsorbed by the body. In the past, people wouldn't have known this was happening and they would've assumed it was a singleton pregnancy all along. As mentioned upthread, if the mother had been pregnant only with the non-viable twin, the pregnancy would've ended in an early miscarriage. Some researchers believe that vanishing twins are more common than previously thought (possibly as common as 1 in 8 pregnancies), but in most pregnancies, they don't know that one of the twins was ever there at all. Most routine scans are done too late to tell as it happens so early in pregnancy.
Vanishing twin isn't genetic. The only thing that is possibly genetic is the tendency to double ovulate. So if your friend's worried about this being genetic, it's not. The only thing she may have inherited is a tendency to double ovulate, but given that many twin pregnancies are due to random factors and not genetics, you cannot say with any certainty that she may have inherited the tendency to double ovulate.