Marie (Countess Marie Larisch) had a few Habsburg archduke cousins but I suspect that the one mentioned here was Rudolph, the only one that was her first cousin.
Rudolph (born in 1858, the same year as Marie) was heir to the Austrian-Hungarian Empire. Rudolph was the son of Marie's aunt Elisabeth, the youngest sister of Marie's father. Elisabeth was married to Emperor Franz-Joseph. In May 1881, Crown Prince Rudolph married Princess Stephanie of Belgium. While Rudolph had numerous affairs, his last was with the 17 year old Baroness Mary Vestera. In January 1889 the Crown Prince and Mary Vestera were found dead, an apparent murder-suicide, at Rudolph's hunting lodge at Mayerling (the name by which the scandal is known).
The other archduke cousin of Marie's that is usually thought of in regard to The Waste Land was Franz-Ferninand, a more distant cousin and 5 years younger than Marie. Upon Rudolph's death Franz-Joseph's younger brother, Carl Ludwig, became next in the line of succession. He renounced his claim to the throne and his son Franz-Ferninand became heir apparent. On the 28th June 1914, Archduke Franz Ferninand and his wife were assassinated at Sarajivo the capital city of Bosnia and Herzegovina by a Serb fanatic working for the Serbian terrorist organization, the Black Hand.
Marie was involved in Rudolph's scandal and it changed her life drastically, in my opinion making Rudolph more likely to be talked about in a conversation with a young American than Franz-Ferninand. Especially, if the conversation between Marie and Eliot took place during Eliot's pre-war trip to Munich in the summer of 1911, as I suspect.
Members of Bavaria's Wittelsbach royal family show up here and there in The Waste Land, either mentioned outright or as possible allusions. See these Miscellanea pages (ordered as they appear):
For much more information on Marie, her aunt, the Empress, her aunt, the Queen, her archduke cousins, her cousin King Ludwig and other royal and imperial family members visit this webpage elsewhere on this website: http://world.std.com/~raparker/exploring/thewasteland/exmarie.html