I am not sure if I understand the expression, but I think "the big thing I have going for me" means "my biggest asset is", but I was wondering if there were any other similar expressions, because I can't find synonyms for expressions like that somehow, there's no dictionary for synonymous idioms.
The big thing I have going for me is perfectly idiomatic.
However, here are some alternatives in the same general vein:
My greatest strength.
My biggest weapon.
My ace in the hole.
My secret advantage.
My hidden talent.
Which you used would depend on exactly what you want to express and how informal you want to sound.
You’re right; “the thing I have going for me” means “my asset”- not sure if I would use “big”. With no adjective, the expression works; “the thing” means essentially “the best thing” (or “the only thing,” depending on context). See examples of usage here: https://idioms.thefreedictionary.com/have+going+for
As for synonyms- “something I have going for me” is like “my advantage” (or “asset”, as you said)- there are many synonyms listed here: https://www.thesaurus.com/browse/advantage