Ideas for a blue martini?

You could use Empress 1908 gin for a regular gin martini. It is blue in color so your martinis will be blue.

Such great ideas, thanks all!!

Buy them some vintage blue martini glasses as a housewarming gift - they can serve a regular martini in them. https://www.chairish.com/product/630913/vintage-cobalt-blue-martini-glasses-set-of-4 (three are plenty of non-vintage ones out there too.)

What about doing this: http://www.jeffreymorgenthaler.com/2006/pegu/
w/ Blue Curacao as the orange liqueur, and no Angostura bitters?

Classic Martini, add a drop of blue food coloring.
Happy New Year!🎊
BB

Blue is hard to find in nature, but I liked the already stated ideas of using blueberries/red cabbage to create a blue base. I'd go one step further and suggest creating a house liquor that can be used, like curacao, to color drinks; but this way you will be able to control the amount of sweetness involved.

Because alcohol is slightly basic it will hopefully prevent things that appear blue from becoming purple. If you don't mind doing some sourcing research cornflowers are bright blue and could be used to create something with a delicate flavor.

I've made some liqueurs from/using this book: https://www.amazon.com/Homemade-Liqueurs-Infused-Spirits-Combinations/dp/1612120989
And Serious Eats has a bunch of guidelines for homemade bar creations, you might also consider trying to create a blue bitters or amaro if your family is seriously sweet averse: http://drinks.seriouseats.com/diy-vs-buy

LJ -
There are many recipes for Blue Curacao martini or margarita out there, some less sweet than others.
Ditto with Blue Agave tequila.
Last resort, have them make up what they prefer to drink and serve it in lovely blue (martini?) glasses.

How about making a blueberry puree and adding it to the typical martini ingredients?

Use curacao. It's a blue citrus liqueur. Blue margaritas come to mind using it instead of triple sec. if they don't like margaritas look for other cocktail recipes which use it.

Two thoughts. Blue in name, though not in color: blue cheese-stuffed olives. More work, but more reward: Play around with pickling red cabbage and use that liquid in lieu of a dirty martini's olive brine. Try: 3/4 cup vinegar (or olive, caper brine, etc), 1 cup shredded red cabbage, 3/4 teaspoon salt (if you're using just vinegar, not brine!). Combine in a pot. Boil until the color is super intense (it'll be fuchsia). Strain out the cabbage. Add baking soda in 1/4 teaspoon increments until the color matches the artwork. That's riffing off an article I wrote about naturally-dyed pickled eggs... so it's a rough guess, but I'm hopeful!