I’m not confident at all with the holes in the goban; Amazons could go either way, and I’ve never played Tumbleweed.
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How about this: Each participant suggests a game they wish to play?
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I like this!
Perhaps up to three each to fill it up.
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I was thinking along these lines, but didn’t want to butt in on someone else’s idea until it was something else entirely. 
Perhaps pick games in the TM sequence until enough games are picked? If there’s a finite list to choose from, that could even be expanded to a pick/ban phase.
Regarding who gets to go first in each game, what about taking it in turns by the TM sequence to say “I will play {player 1 | player 2} in game {game}”, until the first player for each game has been decided? This should make up for the first player advantage being greater in some games than others.
(If doing both, one should of course just continue the TM sequence from the pick/ban (or just pick) phase into the first player choosing phase)
For example, it might go:
Pick/Ban Phase:
Samraku picks Arimaa
bugcat picks Tumbleweed
bugcat picks Amazons
Samraku picks TMG
bugcat picks Go with Holes
Samraku picks Shogi
First Player Phase:
Samraku chooses to play first in Go with Holes
bugcat chooses to play first in Shogi
bugcat chooses to play first in Tumbleweed
Samraku chooses to play first in Amazons
Samraku chooses to play first in TMG
bugcat chooses to play first in Arimaa
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An online friend of mine (re-)discovered this historic chess variant game from GDR-times at her home:
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Speaking of random chess variants, I got this as a prize at a quiz one time. Only got to play a couple of games with it though.
http://www.sjgames.com/proteus/
Basically the way it worked was that you have to move one piece and turn/rotate another. The pieces have values (kind of like ordinary chess if you’re trying to figure out who’s winning in material) and you get points for the pieces you capture. There’s no king so it’s about who gets the most points, or you lose if you can’t move either.
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Any more interest in that hexathalon?
I think it’d be most fun if we could get another couple of people, all choose our games, and then hold the tournament a month or so later after we’ve had time to practice.
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Would there be a finite list to pick from? Perhaps games mentioned in this thread which play well with 2 players and have some way to practically play them online? Since a lot of the rarer games have solid correspondence options to choose from, but no good live implementation, perhaps the rules could allow such games to be played as fast correspondence so we don’t have to do like we did with Amazons and use the forum to notify the other party that we had moved on Little Golem. (Not endorsing that site at all; I don’t like the time control options there, but if it’s the only option, a gentleman’s agreement to play in a fast correspondence style would probably suffice)
L19 discussion of other games that can be played on a goban
chapaev
Lee Sedol playing it, under the Korean name alkagi and with less edible pieces.
A hexagonal Go variant in which, as far as I can make, the idea is that liberties cannot be transferred, meaning that stones can’t enhance their liberties.
Single-post 2020 L19 thread about it.
Oh, and a larger thread. The applauded webpage moved to here.
Also, bugcat, would the picking be once for all matches, per match, or a combination?
Once for All Matches
Each player submits a certain number of games which all players will play in each of their matches
Pros
Encourages players to pick their favorite game(s).
Cons
Games per match grow linearly with the number of participants, which would soon become unwieldy.
Per Match
For each pairing players either submit the (three?) games they want to play, or have a pick/ban phase.
Pros
No limit to the number of players practical.
A wider variety of games may be chosen.
Choices may be tailored to the opponent (perhaps even allow players to negotiate? eg "if I could convince you to submit game A* instead of A, are there any of my choices you should rather see be different?).
Cons
May encourage players to choose games they think their opponent will be bad at, thus leading to many 3-3 draws as both players sweep the games they’ve chosen.
May be more overhead for each match than players wish to put up with.
Combination: Split Picks
Players pick one game to be played in every match, and some number (one?) are chosen by the players prior to each individual match.
Pros
Best of both worlds?
Cons
Worst of both worlds?
Combination: Submission Pool
Players submit games (three? four? two? there should be significantly more submissions than may be played in a given match) to be eligible for play. Then, before each match, there is a pick/ban phase with only the submitted games available.
I would propose something like:
A picks
B picks
B bans
A bans
B picks
A picks
A bans
B bans
B picks
A picks
A chooses first/second
B ditto
A ditto
B ditto
B ditto
A ditto
This compromises between putting bans early and causing the eventual picks to be more even matches, and putting the bans late and letting the players play their favorite games.
Pros
Decisions are easier: submit your three favorite obscure games, and then there’s a finite list to choose from in pick/ban.
Creating this finite list sets up well for allowing interesting pick/ban phases.
No practical limit on the number of players.
Limits the degree to which players can just submit games their opponent has never played, since all options are on the table from the outset.
My preference.
Cons
May be a bit complicated for some?
too much chess
From a different angle, how about a multi-game tournament where the games are nested?
To explain what I mean by nested, here is an example:
Back in high school, some people played a game of Risk, where each battle was decided by the result of a game of Stratego, with the Stratego game being handicapped somehow with respect to the forces in the Risk battle that initiated it.
How about a tournament where the large-scale game is something like Risk/Diplomacy, i.e., multiple factions maneuvering armies to battle over control of the world/Europe, but each battle is settled by a game of Go, where the Go game is handicapped according to the sizes of the forces involved.
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That’s a little intense to start off with…
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I’ve had a good time playing that with people from OSR.
(That is, until I got outplayed too much and got salty ;) )
It’s at its best, I think, as a four-player variant on Board Games Arena.
Oh, Quoridor! I always lose and haven’t the faintest idea why. So it must be a good game, but I have only tried it a few times.