Since Dick Grayson had a living relative, why did the judge allow Bruce Wayne to become his guardian?

Well, she did have serious health problems, as I recall.
Maybe she wasn't deemed fit enough to raise the boy.

Just healthy enough to de-gay the Wayne Manor a little bit.

Pretty much. They killed off Alfred and supposedly brought her in for that reason.

I'm assuming she was Bruce's aunt, not Dick's.

No, she was Dick's aunt.

Definitely Dick's aunt.

Is that like a monkey's uncle?

Bob's your Uncle!

I thought Alfred was there the whole time? When did he die?

Detective Comics #328, one issue after the New Look era began. Aunt Harriet appears for the first time at the end of the story:

Oh. You mean the comic books. I was thinking of the TV show.

Maybe she was just called "Aunt" Harriet...

...but was really his drag-queen Uncle Harry, thus making the situation facepalmingly worse?

Aunt Harriet was a nice old lady and everything, but she wasn't terribly bright.

"Beard" Harriet?

Aunt "Beard"?

On television, the character seemed to be functionally retarded or an alcoholic. The third season, where she was only referred to as being "upstairs, in shock" had me leaning towards the alcoholic explanation.

Not that I blame her. If I had a child like Larry Mondello, I'd be hitting the bottle on a daily basis.

She appeared first in the comics, and was quite clever,
and was on the verge of discovering their secret. Bruce
had to trick her to throw her off the track.

This is why I want John Byrne to takeover Batman - to explain these things.

"Aunt Harriet- Chapter One"

Plus, she was in prison when Dick's parent's were killed.