I've been looking on Wikipedia for info on the families in Star Wars. Apparently Bail Antilles was the Senator for Alderaan just before Bail Organa and then Leia was the senator after him. Bail Antilles had two children - a son, Raymus, who is Captain Antilles of the blockade runner in ANH, and a daughter, Breha, who marries Bail Organa. They, as we know, become Leia's adoptive family. It would be cool if Wedge was Raymus' son, I think. Bail is said to have three siblings, which all seem to be girls: Celly, Rouge and Tia (who had a child, I think male, called Nial). So, we have one known grandparent, one uncle, three aunts and two cousins in Leia's adoptive family.
As discussed, Cliegg Lars has one son, Owen, and Shmi Skywalker has one son, Anakin. Owen married Beru Whitesun. There are no noted extensions to these families.
According to Wikipedia, Padme's surname is Naberrie, Amidala being the queen's name, which she was given after election. Her parents are Ruwee and Jobal (born Jobal Thule). Both grandmothers are named, Winama Naberrie and Ryoo Thule. Padme is said to have an older sister, Sola, who married Darred Janren and had two daughters, Ryoo and Pooja. Pooja is said to have become senator for Naboo up until the dissolution of the senate by the Emperor just before BY. She became friends with Leia, neither aware they were, in fact, cousins.
If anyone knows of any further members of these families in the EU, then please post them.