[[błyskać]]

No, I do not think there is any additional meaning here.

Then what could the person have meant with that sentence? "Flashing with the eyes" seems like "blinking"...

Ah! błyskać oczami/oczyma surely can mean "to blink". But I think the verb is not used in this meaning outside that phrase.

But this is not what is written here. It says "[it] flashes/twinkles from [their] eyes". I need more context to understand this, but this seems like some kind of figurative phrase, related to the "the eyes are the windows to the soul" metaphor.

It was part of an internet chat that I happened to see:

<person1> przekleństwo
<person1> najpopularniejszy znak interpunkcyjny w Polsce
<person1> >.>
<person2> faktycznie popisałem się językiem...
<person1> ... i kulturą :P
<person2> linuxowiec?
<person1> Windows
* person1 błyska z oczu

Reads like bash.

I guess it means "[person2] gives a terrifying look", probably some allusion to the second definition I wrote, so more literally it would be "[person2] strikes a lightning from [their] eyes", but it is really hard to tell. Are you sure you got people's names right?