As I explore this connection, be warned, this post cannot be written without some spoilers.
Now, there is the possibility that Nolan may have chosen this name due to the one of a handful of modern artists named Robert M. Fischer or Rob Fischer, or similar variations, but despite the fact that these men have crafted some impressive modern art, I don't think that the connection between the character of the young energy heir and these modern artists make sense. The connection would only be that these artists create unique worlds, as we also go into the world of Robert Fischer Jr. in Inception.
Bobby Fischer (pictured top) is of course a great chess master, who captivated the world with his game play, logic, and mastery at a young age. The youth element in both of these characters is certainly a similarity.
I'll admit some potential stretches...but I believe there is something here, consider this in part a brainstorming session.
Black & White
The film character is deemed "the mark" the person whom is supposed to have the inception planted. Granted, there's different theories here of what is going on, but assuming that these dreams are all a part of Fischer jr's mind, then it should be interesting to note that perhaps Nolan uses this imagery in the fact that Fischer's subconscious precept ions are first all in black and later all in white. Perhaps when it is black he is fighting off this "white side" and then in the arctic hospital dream he is suddenly joined the side that is accepting the potential inception.
Disconnect from Fathers
Bobby's father story is a little unique, including later discussions and theories about whether the Father on his birth certificate is actually his true father. In the film Inception there is no doubt who the father is, although where film Robert felt isolated from his father, chess Bobby apparently did as well.
'Chess Fischer' seems to have had a strained or at least unique view of family, particularly in his Anti-American and anti-Jewish comments he made in his lifetime...Bobby's family was American immigrants and his mother was Jewish. While 'Film Fischer' does not critic his family, his relationship with his family is quite strained, and the ultimate goal of the inception is for him to disrupt his Father's entire life's mission by breaking a part the company.
Endgame: Rook, Bishop, Pawn
Possibly going way too far, but worth a head spinning if you're interested. Bobby Fischer's famous endgame strategy often involved a Rook (Castle), bishop, and pawns. Now, Ellen Page's character (Ariadne) used a chess piece, a Bishop, for her totem...is there any high castle and Page's assistance (Bishop) that plays a significant role in an endgame in the film? Yes, indeed.
Bobby Fischer died in exile in Iceland. The character Robert Fischer dies in the Ice/Snow dream as well.
Chess: Complexity & Systems
Obviously Chess as a whole serves as a good analogy for some of the action of what is going on. There are rules and order, pieces that move diagonal, straight, or one space at a time. This film creates a system of order as well and to allude to this with a character named Robert Fischer Jr. is simply fun and playful.