Back in 1997 this song was requested 4 times by the same person (chenry1@lcc.net ) in this and 3 other threads:

Big Brother?
1965 ERA Big Brother
Around 1965 Big Brother

This year, someone made a similar request in another forum, and quoted the following lines:

Freckle-faced angel with big brown eyes,
She asked me over for tea and mud pies,
Followed me around and got in my hair,
Wherever I looked, she be there.

Big brother, take me with you.
Wherever you are going, I want to go too.
I think it is funny the things that you do.
Big brother, how I love you.

Went to the fair when it came to town
Rides and cotton candy kept us all day,
And when we got home, she would say,
Mother, there's no need to worry and fuss
My big brother will take care of us.

I also found that 4 country music artists made recordings of a song called BIG BROTHER in the right time frame, but I have not found any lyrics or sound files, so I'm unable to confirm that this is the song that is wanted.

Tex Ritter, 1964
Dickey Lee, 1965
Tompall Glaser and the Glaser Brothers, 1967
Johnny Paycheck, 1967