As several people have already stated, in both speech and writing, the only thing that matters is how the particular writer/speaker would pronounce the sound that follows the indefinite article. If he or she would pronounce that sound as a vowel, it should be 'an', and if as a consonant, it should be 'a'.
Now we apply that rule to your case. In American English as well as in standard UK English, the 'H' in HSC is pronounced 'aitch'. Let's assume that this is how you would pronounce it, too. 'Aitch' begins with a vowel sound, and so should be preceded by 'an'. Thus, you should write it as
He offered to be an HSC.
However, as Ian MacDonald and tmgr have pointed out (and which this article confirms), in the UK, people increasingly pronounce 'H' as 'haitch'. If you are one of these speakers, then you should write it as
He offered to be a HSC.
Here is how this is explained in the Chicago Manual of Style:
7.33: “A” and “an” before abbreviations, symbols, and numerals
Before an abbreviation, a symbol, or a numeral, the use of a or an depends on (or, conversely, determines) how the term is pronounced. In the first example below, “MS” would be pronounced em ess; in the second, it would be pronounced manuscript. In the last two examples, “007” would be pronounced oh oh seven and double oh seven, respectively.
an MS treatment (a treatment for multiple sclerosis)
a MS in the National Library
an NBC anchor
a CBS anchor
an @ sign
an 800 number
an 007 field (in a library catalog)
a 007-style agent
10.9: “A,” “an,” or “the” preceding an abbreviation
When an abbreviation follows an indefinite article, the choice of a or an is determined by the way the abbreviation would be read aloud. Acronyms are read as words and are rarely preceded by a, an, or the (“member nations of NATO”), except when used adjectivally (“a NATO initiative”; “the NATO meeting”). See 10.2; see also 7.33.
a NATO member
a LOOM parade
an AA meeting
a AA battery (pronounced double A)
an NAACP convention
an NBA coach
an HIV test
an MS symptom (a symptom of multiple sclerosis)
a MS by... (would be read as a manuscript by...)
Initialisms, which are read as a series of letters, are often preceded by a definite article (“member nations of the EU”). Whether to include the article may depend on established usage. For example, one would refer to the NBA and the NAACP, on the one hand, but to W3C, PBS, and NATO, on the other—though all these organizations include the definite article in spelled-out form. If no established usage can be determined, use the definite article if it would be used with the spelled-out form. Some terms, such as DIY (do it yourself), do not ordinarily require a definite article in spelled-out form and therefore do not require one as an initialism.