Origin of term "letting on"

Solution 1:

In "Glossary of supposed Americanisms" by Alfred Langdon Elwyn (1859), I find this:

Let on, to mention. "He never let on," he never told me. An Icelandic word, laeta. (Brockett.) "He never let on to me about it," may be heard not unfrequently in the country.

The "Brockett" referred to appears to be "A Glossary of North Country Words", by John Trotter Brockett, FSA (1846) but I can't find the reference to "let on" or "laeta" in the samples that Google provides.

The first place I've found it used with the implication of "lying by omission", as in your example, is in a story called "Little Fairly" by Samuel Lover, Esq. R.H.A. in "The Dublin University magazine", Volume 1 (1833)

"Well there's the scales for you," says big Fairly, and away the little chap wint to weigh his goold (as he let on) as he did before.