Nobody?
Bearing in mind that I've never heard the term before and only googled it ten minutes ago, that's not quite true.
Intergenerational income/earnings elasticity is apparently "a statistical correlation between parent’s and children’s economic standings". You calculate it by saying that the income of a child (once grown) is a function of the income of their primary breadwinner parent and finding the best-fit parameters for ln(I_child) = A + B*ln(I_adult). B is the intergenerational income elasticity.
For example, for parental incomes of [1 2 3 4 5] and child incomes of [2 3 4 5 6], I get an elasticity of .68. If I flip that around so that the child incomes are lower than the parent incomes I get an elasticity of 1.46, so it doesn't even have to be between 0 and 1, which guarantees that this can't be interpreted as "what percentage of your income is due to your parents".
What that .68 means is that the percentage difference between two children is on average (for some sense of 'average') .68 times the percentage difference between their parents. So the second child makes 50% more than the first child while the second parent makes 100% more than the first parent (children's difference is .5 times parents' difference). The fifth child makes 20% more than the fourth child while the fifth parent makes 25% more than the fourth parent (children's difference is .8 times the parents' difference). This elasticity is a measure of how far apart children are relative to how far apart their parents were, all relative to the overall size of the economy that each was earning in. Put another way, it says "if my parents had made X% more money, I would on average be making B*X% more money".
So, to be clear, an elasticity of .5 means that, on average, if your dad made twice as much money as my dad, you ought to be making 50% more money than me.
It's also only as good as the model used to generate it (I've left out some corrections used to adjust for age differences, for example), and it's an on-average kind of statistic such that there will be lots of pairs of children who end up closer together or farther apart than their parents' positions would suggest. You can even see from my example that it could be wrong for particular income pairs even in an on-average sense - it's an average of average differences, kind of.