As @FumbleFingers points out, part of the issue with your first example is the asymmetry of your delimiters. "The youngest of us" is the subordinate clause that should be removable while leaving the sentence intact. That being the case, you should use the same delimiter either side.

Do remember that, unlike some languages such as German, English does not have hard and fast rules about the placement of commas in every possible situation. From a pragmatic stance, they are there to increase readability. In your example, I would contend that excessive punctuation is making the sentence harder to read and understand: I would simply opt for:

I drove my brother Billy, the youngest of us, to the store.

Your question still stands, though, for longer or more complex sentences. At this point, a "hierarchy of punctuation" is a useful concept. A well-accepted example of this is if you are constructing a list of complex items, each of which contains commas. You maintain commas within list items, and use semicolons as delimiters:

I keep only three things on my desk: a pen, as I can't stand typing; a photo of my son, who now lives in Australia and whom I haven't seen in years; and a seashell from the beach where I met my wife.

In the same way, you can use emdashes as your "outer commas" in a clause, and your commas for the inner ones.

I have always held that emdashes are more emphatic than commas - others seem to agree: http://www.thepunctuationguide.com/em-dash.html

In conclusion, though, don't overuse your delimiters: more often than not, you can make do with fewer; if not, do also try looking for a more elegant way of constructing your sentence.

As @FumbleFingers points out, part of the issue with your first example is the asymmetry of your delimiters. "The youngest of us" is the subordinate phrase that should be removable while leaving the sentence intact. That being the case, you should use the same delimiter either side.

Do remember that, unlike some languages such as German, English does not have hard and fast rules about the placement of commas in every possible situation. From a pragmatic stance, they are there to increase readability. In your example, I would contend that excessive punctuation is making the sentence harder to read and understand: I would simply opt for:

I drove my brother Billy, the youngest of us, to the store.

Your question still stands, though, for longer or more complex sentences. At this point, a "hierarchy of punctuation" is a useful concept. A well-accepted example of this is if you are constructing a list of complex items, each of which contains commas. You maintain commas within list items, and use semicolons as delimiters:

I keep only three things on my desk: a pen, as I can't stand typing; a photo of my son, who now lives in Australia and whom I haven't seen in years; and a seashell from the beach where I met my wife.

In the same way, you can use emdashes as your "outer commas" in a clause, and your commas for the inner ones.

I have always held that emdashes are more emphatic than commas - others seem to agree: http://www.thepunctuationguide.com/em-dash.html

In conclusion, though, don't overuse your delimiters: more often than not, you can make do with fewer; if not, do also try looking for a more elegant way of constructing your sentence.