What's your reason for posting this in the folder for Békés county?
The gazetteers don't have anything like Beco (beh-tso). The closest is Becsó (beh-cho) in Zólyom county, now Sebedin-Bečov in Slovakia.
The only way you're going to get anywhere researching your ancestors' lives in Hungary is to find out an exact place of birth for them. I suggest looking into their naturalizations, because the paperwork associated with that should have birthplaces and parents' names. There are others on these forums who know more about that than I do. (I've never needed to look into such things personally, because my parents were the immigrants.)