Learn English – Where did the phrase “you’re it” come from in tag

According to World Wide Words, it originates from firemen doing speed competitions without carrying water.

The term run, more fully fire run, has for at least the past century been used by local fire departments in the USA for a call-out to the site of a fire. It was once common for fire departments or volunteer hose companies to give exhibitions of their prowess at carnivals or similar events. [...] These competitions had fairly standard rules, of which several examples appear in the press of this period, such as in the Olean Democrat of 2 August 1888: “Not less than fifteen or more than seventeen men to each company. Dry run, standing start, each team to be allowed one trial; cart to carry 350 feet of hose in 50 foot lengths ...”.

These reports show that a dry run in the jargon of the fire service at this period was one that didn’t involve the use of water, as opposed to a wet run that did. In some competitions there was a specific class for the latter, one of which was reported in the Salem Daily News for 6 July 1896: “The wet run was made by the Fulton hook and ladder company and the Deluge hose company. The run was made east in Main street to Fawcett’s store where the ladders were raised to the top of the building. The hose company attached [its] hose to a fire plug and ascending the ladder gave a fine exhibition.”

It’s clear that the idea of a dry run being a rehearsal would very readily follow from the jargon usage, though it first appears in print only much later.

The first references to welcome are found in Beowolf.

By 1300, “welcome” was being used more loosely to describe something acceptable, pleasurable, freely permitted, or cordially invited.

From Othello: (circa 1603):

Lodovico: Madam, good night; I humbly thank your ladyship.
Desdemona: Your honour is most welcome.

From the linked source:

A reader found an earlier citation in The House by the Churchyard, an 1863 novel by the Irish writer Sheridan Le Fanu: “ ‘I thank ye again, sir.’ ‘You’re welcome, my honey,’ rejoined Toole, affectionately.”

I don't find it at all mysterious. I imagine it like

You have done well to come to me; I am pleased to do it.

I was taught to say it. In French, they say avec plaisir (with pleasure) or de rien (it's nothing), in Spanish, con gusto (with pleasure) or de nada (it's nothing.) *

(Source)

*_{In the part of Quebec where my parents were born, "Bienvenu" (you're welcome) was the response to "thank you". I actually never heard "de rien" until I was in France, although I don't know what the answer is in Montreal.}