Special thanks to raziel969 for helping me with a few (probably about half) of these.

Cities and Towns

Nibelheim

Nibelheim originates from Nifleheim, the world of the dead, in Norse mythology. In FFVII, Nibelheim was once burned, but Shinra hired people to cover it up, and make it seem like nothing had happened. Therefore, that is how it got the Nibelheim, because it was burned, and was dead.

Midgar

This term originates from Midgard, a term from Norse mythology meaning the middle world, the land of human beings, which was guarded by a serpent. As you know, many of the people live in Midgar, therefore giving it the title land of humans.

Costa Del Sol

This one is rather easy. Costa del Sol means Coast of the Sun in Spanish. The town lies on the coast, and is full of sun and beaches.

Summons

Odin

Odin was the king of all gods, and was also the god of war and wisdom, in Norse mythology. He rode a 6-legged horse, had 2 ravens sitting upon his shoulders, and 2 hounds at his feet. The 6-legged horse is carried through to FFVII, but the other traits dissappear.

Knights of the Round

ANY FF fan should know this one, but I'll tell you anyway. During King Arthur's reign, there were 13 knights, the best knights of all time, that always gathered around a round table. Hence, the EXTREME power of the Knights of the Round.

Shiva

Shiva originates from the Hindu religion. She's the god of war, usually shown with several arms and heads, and is sometimes called "The Destroyer". While FFVII shows her with ice attributes, her true myth shows her with multiple attributes. Square's decision on ice probably originates from the temperatures around India, the prime location of Hinduism, especially in the Himalaya mountain range.

Phoenix

Phoenix was a bird of fire that rose out of its ashes of death every 500 years. That's probably why he is used to heal characters in FFVII.

Ifrit

Ifrit more than likely comes from "Efreet", a fire djinn, hence his elemental attribute.

Bahamut

Bahamut was the king of all dragons in several myths, and is even used in Dungeons and Dragons. Also, he was a giant fish in some other myth, but I highly doubt they decided upon him from that...

Leviathan

Leviathan originates from the bible. It was a gigantic serpent fish that would usually destroy ships. It means "twisted animal" in Hebrew, which ties into the game in which Leviathan usually appears coiled like a snake.

Hades

Hades is another word for hell, from Greek mythology, hence the reason when you summon him it looks rather dark and evil.

Titan

Yet again from Greek mythology, titans were giants that rose out of the Earth. It could also be from Homer's Odyssey, in which the Titans were giant beings, encountered by Ulysses on his famous ten-year Odyssey returning home from the Trojan War.

Ramuh

It could be Rama, the hero of an epic Indian poem, the Ramayana. He is handsome, brave, and a model individual. After many trials and tribulations, he becomes king, and it is revealed that Rama is actually the god Vishnu in human form. However, I could've sworn that he was the god of lightning in some myth...

People

Vincent Valentine

Vincent's last name originates from St. Valentine, the saint of love. Also, there's some other things, such as his clothes: red, which can mean life and death, or love, and black, which means darkness, hate, or sin. Also, his birthday sort of plays into it, also: October being the month that holds Halloween, the day that was originally intended to celebrate death, and the date being 13. 6 is the number of hell, while 7 is the number of heaven. Put them together, and you've got 13, the number representing the imbalance between the two. That all ties in to the Lucrecia sidestory.

Cait Sith

This little cat comes from the Irish language. In Irish, it's "kett shih", which means "fairy cat". As in FFVII, they usually enjoy telling fortunes.

Sephiroth

Here's the background guy. Lots of different possible origins, so here they are: Sephiroth means tree of life, the symbol which can be seen upon the belt that Sephiroth wears. In Jewish religion all beings must travel the Sephiroth or Tree of life, overcoming it's many trials, and eventually, when they overcome such trials, they can reach immortality. Hence, they can become "gods" (or supreme beings). Sephiroth can therefore mean searching for god, or godhood, as Sephiroth tried to do in the game. Also, here's another: It's Hebrew for 'numbers' and is directly related to Arabic 'sifra', French 'chiffre', German 'Ziffer', and English 'cipher'. (Sanskrit 'shunya' meaning 'void', originally.) In Hebrew (among other alphabets), each letter has a numerical value, and the art of adding the letters in a word or phrase to get a certain value is called 'gematria'. In Hebrew, the word Sephiroth has the value of 756; in Greek, 894. Here are the ten important Sephiroth:
1 - Kether, the Crown
2 - Chokmah, Wisdom
3 - Binah, Understanding
4 - Geduldah, Greatness
5 - Geburah, Strength
6 - Tiphemeth, Beauty
7 - Netzach, Firmness and Victory
8 - Hod, Splendor
9 - Yesod, Foundation
10 - Malkuth, the Kingdom
I'm not quite sure which one it is (it may even be both), but as for now, I'd go with the first one.

Hojo

In Japanese, the name Hojo is written with two kanji letters, the first meaning 'treasure', and the second meaning 'reason'. Therefore, this applies to Hojo's love for science or treasuring of Sephiroth, and what Hojo believes as 'reason' being the most valuable thing to him.

Rufus

Rufus in Latin means "red-haired", though I don't believe that ties in anywhere into the character. If anything, I think Rufus was named after Rufus King, a Federalist during the American Revolution. Just like Rufus, he had been appointed as a strong political leader at an early age (29, to be exact). Also, Rufus King played a strong role in the expansion of cities in the West. FFVII's Rufus also wanted to expand towns, though by putting Mako reactors in them.

Biggs

Biggs came from Star Wars; he was a leader of the rebel alliance in the Empire.

Wedge

Wedge was also from Star Wars; he played nearly the same role as Biggs. Hence the reason why those two are usually always partners.

The Cetra

The name for Cetra probably comes from the Latin "cetera", which means "others", hence the reason the Cetra are usually counted as another race altogether.

Shinra Corp.

More commonly known as Silla in Japan, Shinra was one of three kingdoms that occupied the southeast tip of Korea during the 4th to 7th century. In the game, Shinra was the major "kingdom" (or reigning power) of the world of FFVII.

Weapons

Masamune

Masamune has a few possible origins. Here's one: a legendary blade said to be created by a great Japanese smith. Some tales say that the man had to shed his own blood in sacrifice to create the masamune, which is thus forth cursed. The blade therefore was never able to be weilded in battle, for it's power would consume the men that bore it. The blade is said to have a thirst for blood, death, and destruction, and it seems to draw such things toward it. Hence, why there may have been no blood seen when Sephiroth impaled Aeris, seeing as the blade took her lifeblood up into itself. The other goes like this: there was a feudal lord named "Masamune Date" (Date is the family name) who ruled Rikuzen (area near Sendai) around the early 1600's. He lost one of his eyes due to sickness in childhood, and he was nicknamed "Dokuganryu" (dokuganryu means "one-eyed dragon" in Japanese). Another story says that there was a blacksmith named "Masamune", and his sword [also named Masamune (though with a wide blade instead of a thin one as in the game)] was so sharp that you could drive it into the ground in a lake, and leaves that floated by would float around the Masamune due to its holy power. Go for the first origin, I believe it's closer to what Square was going for, though the third one makes more sense when in conjunction with the murasame (considering the fact that the two blades are opposite, and the wielders in the game have many differences).

Murasame

The kanji characters for this blade are "town" (mura) and "rain" (ame). Murasame is used to define a rain that rains in small area. Thus, it could mean murasame can cause "quick blood rain". In another story, the murasame is cursed. Though some leaves floating down the river will avoid the masamune, they float right into the murasame and get split in two.

Rune Blade

Runes were the letters used by Icelanders and other Nordic people over 1000 years ago. They were believed to have magical properties, and supposedly were invented by Odin himself.

Longinus/Spirit Lance

This spear is just known as the Spirit Lance in the US version, but it still has some meaning. Here it is: it was the weapon that pierced Christ's side after they took him down from the cross, and also the name of the Roman centurion who used it. Some people say that whoever used it would be granted immortality, also.

Ragnarok Sword

Norse mythology's "Destruction of the Powers"; the climactic battle between the monsters and the Gods. In the game, you receive the Ragnarok after defeating the monster "Proud Clod", one of the final battles. Because of your amazing power in the end of the game, you could be seen as "Gods". Hence the name of the sword after defeating the final "monster" (not including the part-humans).

Winchester

This one's easy.. A winchester is a type of gun. Vincent uses guns, and that's why one of his weapons was named that.

Creatures

Midgar Zolom

Do you remember that annoying snake that guarded the entrance to the Mythril Mine? The one Sephiroth impaled on a tree? It's named the Midgard Serpent, or originally Midgardsormr. In Norse Myth, Midgard was thought to be a huge fortress encircled by a great serpent which lives beneath the sea (which could explain why the serpent is shown as being underneath the sand instead of on top of it). In Japanese, r's and l's are barely distinguished between, so that Midgardsormr could be read as Midgar Zolom, as it is called in the game.