Do elves have surnames/family name? For instance what is the surname or famous elves like Galadriel, Arwen, Legolas, ...
Just like modern humans, the various peoples of Middle-Earth all have different naming conventions. In fact the only people in Middle Earth to have family names analogous to the common (but not universal) modern format of Firstname Lastname are the Hobbits of The Shire (Bilbo Baggins, Peregrin Took) and presumably also of Bree (the Underhills) and the Men of Bree (Barliman Butterman).
So, NO, none of the Elves (Galadriel, Arwen or Legolas) had "family names" as we understand them. However, they did have second names and epithets that were often used in the same FirstName LastName format as the usual modern usage of Surnames.
The naming tradition of the Valinorian Elves - such as Galadriel and the elves of The Silmarillion, the essë, gave an Elf three different names at various times of the lives.
Essi: chosen by the father and often based on the father's own name or derived from old legends. Given at or near birth. Galadriel's father-name was Artanis meaning "Noble Woman" in Noldolin Quenya. Maedhros's father-name was Nelyafinwë (literally Finwë the Third).
Amilessë: chosen by the mother and often prophetic in nature. Given sometimes at or near birth but often bestowed years later. Galadriel's mother-name was Nerwen meaning "Man-Maiden" in Noldolin Quenya (referring to her height and bold character). Maedhros's mother-name was Maitimo (the "Well-Shaped One" was known to be good-looking)
Epessë: often acquired by an Elf as an honorific later in life. Galadriel's epessë was Alatáriel meaning "Maiden Crowned with Radiant Garland" in Telerin Quenya, given to her by Celeborn, her husband. Maedhros's epessë was Russandol ("Copper-Top" - he was a rare red-headed elf).
When these Valinorian elves made their way to Middle-Earth, where Sindarin was the common elven tongue, they adopted Sindarinized versions of their names. So Alatáriel became Galadriel and Maitimo / Russandol became Maedhros.
Both Maedhros and Galadriel, like most other elves in the era of the Silmarillion are referred to with those single names only, without their names being doubled in the FN LN format anywhere in the canon, though Maedhros was sometimes called "Maedhros the Tall".
On the other hand, one of the exceptions was Galadriel's brother Finrod. He was known as Finrod Felagund - the former part being the Sindarized version of his father-name and the latter part being the Sindarized version of an epessë given to him by Dwarves - Felakgundu meaning "Hewer of Caves".
We do not know if Arwen was her father-name or mother-name, or if the later day elves in Rivendell even still followed this tradition at the time of her birth, but she does have an epessë, Undomiel and she was often referred formally as Arwen Undomiel. As such the name Undomiel serves as part of a double name in the format of FirstName LastName, but is not a true surname since it is unique to her and does not refer to the family.
Legolas had no ancestry from the Valinorian Elves and we do not have much information on the naming customs of the Sindar and the Silvan elves, among whose numbers Legolas would be counted. He is almost always referred to as Legolas, in the singular, except for the one time when Gandalf calls him Legolas Greenleaf, which again serves as adequate example of FirstName LastName format. However, "Greenleaf" is simply the literal English translation of "Legolas", so again, it cannot be counted as a true surname.