I say "Congress", but I gather this would work for any high-ranking official, public or private, just short of the President.  Or maybe even him… or her…, depending upon press coverage.  But to be its most effective, you would use it to contact every single senator and representative in Congress at the same time.

And the reason I'm putting it on this site is because, to the best of my knowledge, this has never been done.  So I guess it's an official 'secret' until somebody does, hence its inclusion.  If done correctly, it's only going to work once.
 

The idea is relatively simple.  It's the size that'll create the stir and thus grab the staff's attention, and eventually the bigwig's attention.  And the media's attention, should they accidentally happen to be walking down the hall with a camera crew just as the envelopes arrive.

To wit:

1.  Write them your meaningful-but-polite letter, enclose it in an envelope, address it, stick a stamp on it and send it.

2.  Except make everything 10 times larger than normal.

3.  Everything.

The writing paper would be the size of a small wall.  The fonts would be proportionately gigantic.  Ditto the envelope and stamp.  You'd put cancellation marks on the stamp, so it would look real, but actually send it via FedEx.  I'm picturing the envelope roughly 3 feet tall and 5 feet wide.

Picture the scene.  The FedEx guys shows up with a "Got a letter for ya!", lugs the damn thing in, everyone clusters around, and eventually the ruckus gets the attention of the senator and his immediate aides in the back room.  "Come here, you've got to see this!" one of the girls calls.

The senator walks to the front room as one of the staff opens the gigantic envelope.

And there you go.  You've got the senator's attention for about one minute.  Novelty or not, he's a busy guy.  This is it.  Ball's in your court.
 

Acquiring the paper for the letter and the envelope would probably require an actual paper factory and might cost a pretty penny, but at least the order's uncomplicated.

As far as the lettering goes, there are a few different routes you could take, from printing them out and gluing them on the master page, to using those rub-on letters, to swapping a case of chilled beer with some guy who works at a printing shop for a secret Saturday night run.

As for the stamp, standard posterboard would be perfect as its thickness would mimic the thickness of a real stamp.  You'd print the stamp in sections and then carefully glue them together, then take some colored pens and make sure the joined edges are as invisible as possible.  If you have access to a hi-res scanner, you could scan an actual canceled post and use that, but that's a pretty big leap, size-wise, and it might be blurry as hell by the time it's enlarged.  You'll probably do better digging up a hi-res pic of the stamp and putting on the canceled marks with Photoshop.

About all I can say at this point is, if you actually do it, for gawd's sake don't screw it up.  National press-wise, I figure this is good for two shots.  Sending one to a single senator's office might make for a photo in one of the local rags, possibly the evening news.

But sending one to all 400-plus congresscritters would definitely grab the headlines.  And, of course, presumably your fervent message-slash-request-slash-demand-slash-grievance would finally be aired on national TV.  Make sure you have a good agent on tap so you can secure the movie and book rights as soon as you become the first common citizen in American history to garner the attention of every single member of Congress at once.

And, by way of extension, the entire world.