My three year old wanted to play Ticket to Ride because he likes trains, so we worked out a variant together.
Pick a color, grab all the trains, and pick a starting city. Each turn claim a route that has to start from the city you're currently "in". (You're actually moving from city to city, and since the game is supposed to be a race across North America, this is actually more realistic than the real game.) Put down the appropriate number of trains. No cards are used. Announce the city you're moving to.
If the city has a baseball team, say what you think of the team, i.e. "Toronto Blue Jays are ok!" or "Boston Red Sox stink!"
End game: someone runs out of trains and yells "I win!"