By how much does being told what piece to move improve one's Elo? Here is a (rather humorous) game in which a 900 player is told by a much higher rated player only which kind of piece to move (e.g. along the lines of "bishop", not "light square bishop"). 

Is it safe to assume that this assistance would improve the player's game, and presuming so, is there any statistical approximation (or rule of thumb) to estimate the expected improvement in someone's performance (measured by Elo) when given such an advantage? 

E.g. would a 900 player assisted in this way by a 2700 player become a 1500 player? (that's a wild guess for argument's sake)
This chess variant is called hand and brain.

From watching many of these games, I can guess that the only boost achieved is that low rated players get advice on a piece which is hanging.  So given your example, the hand would appear to gain Elo (named after Arpad Elo) just for not leaving pieces en prise.  Another advantage, although not very useful, is that by restricting the number of moves from the usual 30 down to about 8 does increase the shear odds of guessing the correct move.

The downside of this game is that the tactics (which most consider to be 90-99% of chess) is severely restricted.  It's dangerous start a combination or even a simple attack unless the lower rated player understand.  It's also possible that a difference in playing style would adversely effect the choice of moves.  That is, an aggressive player always wants to push forward, and a positional player wants to suffocate to opponent, so each has a different view of the most correct move. 

There currently isn't enough data to give a statistical answer, but I can guess about a curve.  A low rated player could play about 300 lower than the higher rated player with a limit around 1500.  As the lower rated player rating rises, this playing strength average around 100 points below the lower rated player.  As the lower rated player rating reaches the master level, the playing strength almost disappears.

EDIT:
Expanding answer to better explain and suggest research.
I've only seen about 50 hand and brain games, so my data is unofficial and sprace.
My first exposure to hand and brain chess was on twitch.tv watching Anna Rudolf(2200) and Alexandra Botez(2000) play, from their respective channels.
Two channels which often have a pair of players that have played are HashtagChess (2200 and 1800) and CurlyQueenn (both 1900).  (On of the members of CurlyQueen is getting her own channel, so they may stop streaming together.)
I've seen other channels play this game, but they don't play regularly enough to be suggestions.

The chatter shows that they often guess wrongly about what the brain move was.  But, since there is a chat box, you could ask them directly for their opinions.  You could note the games and have computer analyse and compare to their normal games.  (You can tell which site they are playing from and can see their user name, but I'm sure that they will be helpful if you just ask.)

NOTE:  I have some objections to the enforcement of twitch.tv policies, so I won't be able to get this information directly.