I used to teach elementary school and preschool. This list was something I added to during those years.
Note that the Wikipedia page on tag has other variants.
One person is initially it. Whomever is tagged becomes the new it.
One person is initially it and behaves like a monster. Anyone tagged is now also it and should also act like a monster. The last person not it starts the next round as the initial it.
The game is played like Cumulative Tag but all its hold hands and only the two free hands at the ends of "the blob" can tag others.
The game is played like Big Blob, but any blob of four people splits into two blobs of two people.
One person is initially it, all others hide. After it finishes counting or reciting a rhyme while at the base with eyes closed, the action begins. The hiding players try to get back to the base while the it player looks for them. The first person tagged each round is it next round.
The game is played like Hide and Go Seek, but all people tagged become its the next round.
Everyone but one person is it. The not it person hides. Anyone who finds the hiding place also hides there, holding hands. Either the first or last person to find the hiding place is next round's it.
The players form two teams. Each team has people given one cards of each rank (ace, 2, 3, 4, 5, ...) to match the number of teammates. Everyone is it. When a tag happens the two players show each other their cards and the player with lower rank is out of the game until the round ends. The first team to "capture" the other team's ace wins the round. (I've also seen this variant done with Stratego pieces).
One or more players are initially it. Tags by the its freeze people. Any non-it can tag a frozen player to unfreeze. The game ends when all non-its are frozen.
Before the game begins everyone's name is put in hat. Each player draws on name (repeat until no one has their own name). Everyone is it but only tries to tag the person whose name they have. The game ends in set time limit (10 minutes or so) and works best played in a somewhat confined space such as hallways in one floor of a building. In a nighttime variation the tags happen not with touches but with flashlight flashes.
The game is played like Killer, but when you tag someone you get their paper with the corresponding goal of a new person to tag.
The game has a setup like Killer, but instead of names blank pieces of paper are put in hat: only one paper has mark and the person who draws that paper is secretly the initial it. The game is played like Monster except that the its quietly spread throughout the players.
The game has a setup like Monster Killer to select a secret initial it. Instead of tags making a non-it another it, the tags freeze the non-it as in Freeze Tag. Unlike Freeze Tag, being frozen is permanent until the round ends.
The game has a setup like Monster Killer, but two papers distinctly marked. The player who picks the first mark is the secret freeze-it whose tags freeze non-its; the player who picks the second mark is the secret monster-it whose touch turns a frozen player into another freeze-it.
The players form two teams. Each team has certain number of "shots" made from bit of flour wrapped in tissue and taped closed. Everyone is it if they still have "shots". Tags happen by hitting someone with "shot" so it breaks open. Whichever team first finds the other team's hidden "treasure" (a handkerchief or something) wins.
The game is played like Pirates, but each team hides/carries the "treasure" the other team needs to win.
There are two bases, with one it for each base. The its throw a ball back and forth while all other players try to run back and forth from one base to the other. A player tagged with the ball replaces the it who tagged him or her. Runners gain prestige by making many runs back and forth between bases.
Play with fewer frisbees than players. Whomever a frisbee has last touched is obligated to make that frisbee touch someone else.
The game is played like Tag, but a player is safe if he or she sits down and says the name of an item from the game's theme that has not been used before. Common themes are television shows, symphony instruments, famous physicists, etc.