As you're aware, biochemists typically do not use systematic naming for most compounds, preferring instead to use common names and modifying them as needed for related compounds. In the case of amino acids, the names typically are created based on the names of the proteinogenic amino acid names. I do not know of any codified formal rules for this process, but a quick look at the names in use suggests a few guidelines that seem to be followed most of the time. Here's an imcomplete summary.
1: Priority seems to be given to the "R" functional group rather than, for example, chain length. Thus, $\ce{HS-CH2-CH2-CH(-NH2)-CO2}$ is named as "homocysteine" based on the presence of the -SH group rather than as a variant of methionine, which has the same number of carbons and a sulfur atom, but not a sulfhydryl group. Similarly, $\ce{HO-CH2-CH2-CH(-NH2)-CO2}$ is named homoserine rather than considered a reduced form of aspartate, which has the same number of carbons.
2: The prefix "homo" indicates an extra methylene group in the side chain (eg homoserine, homocysteine, homolysine)
3: The prefix "nor" indicates a structural isomer of an alkyl side chain (eg norvaline, norleucine)
4: Aldehydes tend to be named as carboxylic acid derivatives rather than alcohols, so $\ce{OCH-CH2-CH(-NH2)-CO2}$ is aspartate semialdehyde rather than an oxidized homoserine.
5: The suffix -ol designates reduction of the 1-carboxylic acid group to an alcohol.
The sulfonic acid compound you asked about poses an interesting challenge, because it does not have a methyl group on the S, but also is not an -SH group. One could make an argument for naming this compound as either a methionine or homocysteine derivative, but convention seems to favor homocysteine, based on similar compounds such as S-adenosyl homocysteine. Thus it would be "homocysteine sulfonic acid".
As noted in the comments, your aldehyde compound is typically named glutamate-1-semialdehdye to indicate that only one acid group (at the 1 position) is reduced.
