My brother wants new headphones as a New Yer present, he has lost his old .......... (one/ones). If you buy two computer games, you get a third ........(one/ones) free.

In your sentences, you should use "ones" in both cases.

So, it would read:

"My brother wants new headphones as a New Year present, he has lost his old ones."

"If you buy two computer games, you get a third one free."

"One" is used when referring to a single item, while "ones" is used for plural items that have already been mentioned or can be understood from the context.