"Irrational" design in these papers refers to combinatorial mutagenesis, which is put forward as the alternative to "rational" protein design.
Rational design involves using existing information about a protein to choose residues to mutate.
From Nixon et al.:
Alteration of function by rational approaches can be achieved through single-point mutation, exchange of elements of secondary structure, exchange of whole domains, or generation of fusion proteins
Whereas irrational design uses random mutations to gain insight into the functional capacity of specific residues or domains, which may then be further manipulated by rational approaches.
From Tobin et al.:
‘Irrational’ protein design using directed evolution methods relying on entirely random mutagenesis techniques has also had its successes, particularly in cases in which sequential rounds of mutagenesis have been pursued.