Jech, in the book Set Theory, 3rd Millennium edition points the blame to Addison in the historic notes to the first chapter about descriptive set theory (i.e. Borel and Analytic sets).
The definition of the projective hierarchy was given by Luzin in 1927. In 1955 Kleene published his hierarchy of analytical predicates. Several years later, Addison published a paper binding the two and gave the distinction of boldface/lightface hierarchies.
Here is a link and a MR link:
J. W. Addison. “Separation principles in the hierarchies of classical and effective descriptive set theory“. Fundamenta Mathematicae, vol. 46 no. 2 (1958), pp. 123-135.
I found an answer with a lucky search. John Addison claims in his paper “Warsaw 1957: Memories of Mostowski” that he invented the $Pi^n_t$ and $Sigma^n_t$ notation, along with the boldface/lightface distinction. This was in the early 1950s.
You’re right that boldface is not a particularly good way to make a distinction in notation. In particular, typewriters cannot produce bold and photocopiers can reduce the distinction between bold and regular text. A separate convention developed to write a tilde under the $Sigma$ or $Pi$ to denote boldface, particularly in typewritten works.