The title is a reference to the saying "The luck of the Irish".
"And it's a dead heat! They're checking the electron microscope. And the winner is...number 3, in a quantum finish." "No fair! You changed the outcome by measuring it." According to the Observer Effect, it is believed that measuring a particle ultimately causes it to change its position. Professor Farnsworth is arguing that by measuring the race results, the position of the horses (and thus, the outcome of the race) was altered.
Harry Trotter, one of the race horses, is a play on the name of the famous children's book character Harry Potter.
Yancy lights a thin strand of Playdoh to blow open the Ronco safe. In real life, Playdoh is composed of non explosive materials that are safe for kids.
The song at the end was Don't You (Forget About Me), a song from the 1985 movie The Breakfast Club. The producers said this episode was meant to be depressing in order to give the audience a feeling of Fry's dealing with the loss of his 20th century friends and blood relatives.
Coincidentally, the beginning and end of The Breakfast Club use the song, and a plotpoint in the 20th century B-plot is Fry hiding the clover in the soundtrack to the film.
The beat boxer for the breakdance crew was named Noticeably F.A.T. A homage to rapper Notorious B.I.G
When Fry goes to dig up his brother's grave he says, "It's clovering time." which is a reference to The Thing's catchphrase, "It's clobbering time."
When rummaging through Fry's belongings, Yancy pulls a banner out that says "Whitefish". This is a call back to the episode "Mars University" in season one when Fry states that he is a dropout of Coney Island College and shouts, "Go Whitefish!" "Coney Island whitefish" is slang for publicly-discarded used condoms.
When Bender opens the long-enclosed safe, Bart Simpson's head, seemingly from a doll, is seen on a shelf above.
When Fry's father states that the family will need bananas when "the radiation turns us all into monkeys", he climbs the ladder out of his basement/bunker in a way that is reminiscent of Donkey Kong - a famous ape - in the Donkey Kong series of video games.
The title of the episode is most likely a reference to the 2001 movie The Luck of the Irish.
When the gang is in Old New York looking for Fry's clover, there is a brief moment where in the background of the rubble you can see the remains of the Twin Towers, although the attacks of 9/11 destroyed them in real life only 6 months after the episode was aired.