With a single indicator, the unstandardized loading must be fixed to 1 and the error variance must be either fixed to zero (assuming/implying perfect reliability of the indicator--not a very realistic scenario) or to a theoretically meaningful value that corresponds to your "best guess" as to how much error variance there is in the observed variable. The estimate of the error variance can sometimes be derived from reliability estimates (but be careful when using reliability estimates from prior studies as they may not apply to your sample/population).

With known reliability, the error variance for an observed variable (indicator) Y can, according to classical test theory, be computed as

Var(error) = Var(Y)*(1 - reliability).

It is not meaningful to estimate the loading of a single indicator. The unstandardized loading must be fixed to one for identification. The standardized loading will be "estimated" but will trivially correspond to the square root of the reliability coefficient when fixing the error variance according to the formula above. So there is no new information there.

Thank you for sharing your input, Dr Christian Geiser

In SEM, a single indicator latent variable is a latent variable with only one indicator. The factor loading for a single indicator latent variable is equal to the square root of the variance of the indicator. You can find more information on SEM in Helmut Lütkepohl and Markus Krätzig’s book “Structural Vector Autoregressive Modeling and Impulse Responses”.