I think that she was the daughter of one of his closest friends. That friend and his wife died and Barrett took Marline in.

It's seems to me that Barret adopted Marlene for 3 main reasons.

#1 Dyne was his friend. Wouldn't you consider taking in your friends kids if he was killed?

#2 The town Marlene lived in was destroyed. She didn't have anywhere to go really. She knew Barret so it was almost their only choice.

#3 Barret felt constant guilt for arguing in favor for the Mako reactor and subsequent destruction of the town. Adopting Marlene would allow him the feeling of atonement for his transgressions.