What significance does Mike Yanagita have in Fargo?

I agree with the comment that Mike's convincing lying points out to Marge that Jerry may be lying too, and so the second interview with Jerry, where Jerry becomes less and less convincing ("I'm cooperating here") under Marge's polite but insistent questioning.

But I think there is something else. Consider the consecutive scenes where the criminals are banging the TV and yelling obscenities at it, followed by Marge and Norm calmly watching a show where insects prepares for birth by storing food, followed by the phone call from Mike. The next day, she decides to go to the Twin Cities, a four hour drive from Brainerd. Notice how surprised Norm is, and notice how she doesn't tell him about Mike. Does she really have to go to the Twin Cities? Not really, a state trooper was killed, the state police would be all over this investigation. She is going to see Mike, who she knew and liked in high school, and who she may feel would be a better provider for her and her child than the virtually unemployed Norm. The investigation is a convenient cover (and pays the expenses) for the trip.

Of course, when she gets there, she realizes that something is wrong with Mike, she cuts the meeting short, and next morning she calls her girl friend in the Twin Cities to find out. She is surprised to find that everything Mike said was a lie, and he is a mess in general. Which leads her back to Jerry, who is even more of mess.

At the end of the movie, again watching TV in bed, Marge learns that Norm sold a painting to the Postal Service, so some money will be coming. So Marge now feels they are doing pretty well (she may not have felt that before). So the point of the sub-plot - it is better to do the lawful thing, rather than be tempted to circumvent the law - reflects the point of the main plot, which is the same point exaggerated by the complete stupidity of the criminals and the magnitude and violence of their crimes.