Things that are important:
Hex grid movement, obviously
Hot-seat or online play - it's a multiplayer board game, thus a multiplayer video game
Keep track of what you might call "touchdowns" at a specific location, as well as HP and movement speed
AI would be unnecessary, so would sound effects or a matchmaking server. I just want to be able to show this to people.
None of these really factor into what language you might choose.
Flash, because it's ubiquitous.
This is true. There's also a lot work that's recently gone into making it easy to make Flash games (see Flixel and Flashpunk as examples).
Java, because it seems to be the more powerful successor to flash.
C# seems to be an industry standard for video games.
"Industry standard" is a strange term to use here. When you're talking about large console and PC games ("AAA" games), C++ is actually the indsutry standard. If you're talking about Unity games, or tools development, or XNA, then C# is the standard. C# is a great language. However, the reasons I find it great are because of the advanced programming features it offers over Java. This may not be as useful to you.
I'm going to argue against using Unity for a couple of reasons:
You need to purchase a tool to help you make 2D games.
Learning to program with Unity's programming model will leave you with gaps in knowledge about how to structure a program from the top level. It revolves around writing scripts to create components that are then attached to objects. You are shielded from the main loop, which is an important concept to grasp in game programming. Unity's awesome and I love it, but I wouldn't recommend learning to program with it.
Python, because geeks always geek out about how awesome it is.
Python is wonderful (so is Ruby). You can get a lot done quickly, and making a basic 2D game with pygame is very straightforward. However, you run into problems if you want to distribute the game. There are tools, py2exe and py2app, to help you distribute, but I've never had either work 100% reliably.
If you don't intend to be able to distribute your game, then Python is a great way to go. Check out Learn Python the Hard Way and Invent Your Own Computer Games with Python. If you'd like to be able to have people easily download your game for playtesting or to just be able to set it up on a computer other than your own, I'd recommend Flash. C# and Java are nice, but if your primary goal is not to make a computer game you may find it more efficient to use Flash or Python.
Both Flash and Python can do online play, but there may be more readymade tools and libraries available for enabling online multiplayer in Flash games. Be aware that this is not a trivial problem to solve.