Inspired by a local gaming convention in the area that I have attended occasionally in the past, GameStorm.
Most games will tell you which player makes the first move. For example: Hungry Hungry Hipsters - “The player with the least purposeful glasses plays first. Play proceeds counter-clockwise.” or even stranger, Bridges of Shangri-La: “Whoever has last reached the peak of Mount Everest using nothing but blue and white checkered stilts carved from the wood of a Mammoth tree is declared the Starting Player. In case of a tie, the wisest player of the group begins the game.”
It is your move. But real games only, please. 
Not weird, but the rule for Azul is the last player who visited Lisbon. I played it on a cruise where the next stop was Lisbon. Had I played the next night, we would have to figure out who got on the ship last.
The older version of https://boardgamegeek.com/boardgame/12495/fire-and-axe-a-viking-saga (may have been when it was “Viking Fury”)
Decide which Viking will take the first turn (he of the hairiest head).The chosen Viking is called First Player. He gets a victory point.
Munchkin:
Decide who goes first by rolling the dice and arguing about the results and the meaning of this sentence and whether the fact that a word seems to be missing any effect.
Brian
According to the official Cards Against Humanity Rules, the person who most recently pooped goes first. They start the game as the first “Card Czar” (the judge for the round). (Unless you are playing with Hugh Jackman, who always goes first if he is present).
I can’t remember the games, but I’ve seen “the player whose birthday is next” and “the youngest player”.
But these other ones are definitely weirder.
The Bloody Inn has the players as innkeepers who rob and murder guests, then bury them below the floorboards of their inn. The player with the most coins in their pocket starts the game.
Pandemic: The player who was most recently sick goes first.
Forest Shuffle: The player who most recently took a walk in a forest goes first.
Those are all fairly topical, though, not terribly weird.
Pillars of the Earth has the player who most recently visited a cathedral going first. In our group that would always be the same person, so we ignore it.
Quite a few games have a weird or thematic start player also have a “or choose randomly”
In my main group we use an app or drop player pieces. In another group I always go first as I am usually teaching.
The rule for Railroad Dice is “Choose a start player according to your preferred method (the player who owns the most railroad games)”
Brian
My copy of Pandemic says the players receive ‘city’ cards from a shuffled deck.
Then the player with the highest city population goes first.
There are over 3100 Cathedrals in the world. How close are you to one?
The rules for Exploding Kittens say that you can determine who goes first however you want, but then they give a few suggestions. One of them is the player with the shortest spleen.
I’m within a half-hour bike ride of at least two, but I don’t think I’ve been inside either, or in any other meaningful sense “visited”, since before Covid.
I mean how much of an advantage is it to go first in that game and how competitive are you? It’s an almost two hour drive to the nearest cathedral to me but I’m not too far from it a few times a year and could make a side trip.
Then again, board games are usually played in person. If a group of people are all sitting around a table, they’re, at least at that time, all equally far from a cathedral, and could equally easily make a side trip there (at least, on that trip, if some of them are coming from elsewhere).
It should be noted that, even if everyone at the table poops with the same frequency, this will still consistently favor some players over others. If, for instance, both Alice and Bob poop once per day, but Alice does it as soon as she gets up, but Bob does it right after breakfast, then Bob will almost always be the more recent pooper of the two.
What about Carol and Ted? 
Youngest player goes first is the rule in Candyland.
Assuming that you’re looking for whimsical rules, not the weirdest (if “the weirdest” the OP probably wins), there’s Cascadia, a lovely little ecosystem-building game based on the Pacific Northwest. The game revolves around bears, hawks, salmon, foxes, and elk, and whoever most recently saw one of these animals goes first.
For Arschmallows, the player who most recently ate candy goes first.
I can imagine in some circles this rule could lead to bitter arguments. “OK, we’ll count the Episcopalian and Ukrainian Orthodox cathedrals, but that Celestial Church of Christ one is just a storefront, and their claim to apostolic succession is dubious at best!”
I have a definite advantage when playing Cards Against Humanity. I poop three times a day most days, sometimes as many as five. This is normal. A one-poop day is atypical for me. Yay me.
“Sure, I’ll play… right after I hit the bathroom first.”