The phrase "his eyes shone" is correct and can be used in written English. You can use this phrase to describe someone's expression, or to emphasize that they have a particular emotion such as excitement or joy. For example, "When he heard the news, his eyes shone with anticipation of the upcoming adventure."
Mr Kim knew this was my final full day in Cambodia, and after I had waved a long and lingering farewell to Sopheng and Sokoun at the CBT building, he looked at me and asked, "When do you have to be back at Siem Reap?" Not until dinnertime, I said, and his eyes shone.
His eyes shone at the idea.
His eyes shone from slits cut deep into his face.
Round his heart resentment welled, and his eyes shone out like licking fire.
His eyes shone behind his glasses when I said the words.
"Are you looking forward to the conference?" His eyes shone and he nodded a yes.
"Then … why are you here?" Her eyes shone.
Osama's eyes shone.
A pretty face, sharply focussed, very dark-brown eyes, shone in the thing's green light.
When Helen mentioned that the house had been owned by a doctor, who kept a log which she had found in the attic, Lydia's eyes shone.
A disciplined gleam shone in his eyes.
Being a terminologist, I care about word choice. Ludwig simply helps me pick the best words for any translation. Five stars!
Maria Pia Montoro
Terminologist and Q/A Analyst @ Translation Centre for the Bodies of the European Union