How to store one-byte integer in PostgreSQL?

No, there is no 1-byte integer in the standard distribution of Postgres. All built-in numeric types of standard Postgres occupy 2 or more bytes.

Extension pguint

But yes, there is the extension pguint, maintained by Peter Eisentraut, one of the Postgres core developers. It's not part of the standard distribution:

In addition to various unsigned integer types, it also provides the 1-byte integer you are looking for:

int1 (signed 8-bit integer)
uint1 (unsigned 8-bit integer)
uint2 (unsigned 16-bit integer)
uint4 (unsigned 32-bit integer)
uint8 (unsigned 64-bit integer)

Be sure to read the chapter "Discussion" at the linked site, explaining possible complications. You need to exercise care with type casts and numeric literals when introducing more integer types ...

Workaround

A possible, simple workaround would be to encode 1-byte integer values as "char" (with double-quotes!), an "internal" simplistic 1-character type, which actually uses a single byte of storage, byte values of a signed 1-byte integer, the upper half represented as ASCII characters.

You can encode values in the range of -128 to 127. Demo:

SELECT i
     , i::"char"
     , i::"char"::int
FROM   generate_series(-128,127) i;

There are several characters not meant for display. So encode before you store and decode before you display ...

Remember: "char" is an "internal" type intended for simple and cheap enumeration. Not officially designed for what we are doing here, and not portable to other RDBMS. There are no guarantees by the Postgres project. But since "char" is used all over the system catalogs, the type is not going to change.

_{Initially, I had assumed unsigned 1-byte integer range (0 to 255) and used text as stepping stone. But that only works for numbers 1 - 127. Instead, use signed integer range (-128 to 127) and cast between "char" and integer directly.}