According to Jeff Miller's Earliest Known Uses of Some of the Words of Mathematics, the term kernel was first used in this meaning by Pontryagin (in translation by Lehmer) in the slightly broader context of group homomorphisms. It doesn't say why Pontryagin or Lehmer chose that particular word though.
matrices - Word origin / meaning of 'kernel' in linear algebra ...
78 A semicolon is used to separate variables from parameters. Quite often, the terms variables and parameters are used interchangeably, but with a semicolon the meaning is that we are defining a function of the parameters that returns a function of the variables.