The word "Abiathar" in the text may go all the way back to Jesus and it's entirely possible he misspoke.
As Bruce Alderman's answer ably points out, we can never know the exact words that Jesus spoke as there were no recording devices at the time. Therefore, we must rely on the people who heard Jesus' words to remember them until they could be recorded in writing. For a critical 30 or 40 years, the teaching must survive in the form of oral traditions. There is very little evidence for what, if any, changes might have occurred to the teachings in this period.
However, we do know that Jesus' disciples considered Jesus to be a rabbi who spoke with authority (semikhah). The job of a rabbi was to pass on his wisdom to his disciples and the job of the disciples was to remember their master's teaching. Part of the reason Jesus picked twelve men to be his "inner circle" seems to be that he wanted them to have an even deeper understanding of his teaching than the crowds (cf. Mark 4:10-20). These men followed him for three or so years and probably heard the same teachings over and over again. If they had any questions, they would likely have been expected to ask Jesus privately.
(An argument might be made that this particular teaching was a one-off since it related to a very specific event: the disciples gleaning on the Sabbath. But it's also possible the disciples made a regular practice of picking grain on the Sabbath and used Jesus' answer multiple times. I don't think it really matters to my argument except tangentially.)
Now the author of Mark assembled some of the oral history that was preserved and wrote it down. It seems likely that he wrote the stories about Jesus in the format of an ancient biography in order to transmit them to gentile Christians. Around the same time or a little later, Plutarch wrote his Lives which was popular in the Greek-speaking world and share some literary features with Mark's gospel. (Another possibility is that Mark realized the generation of Apostles was dying out and that the oral tradition ought to be preserved in writing.)
Here is the parallel passages with unique matterial in bold and omitted words marked (...) Mark 2:26 (ESV):
how he entered the house of God, in the time of Abiathar the high priest, and ate the bread of the Presence, which it is not lawful for any but the priests to eat, and also gave it to those who were with him?”
Matthew 12:4 (ESV):
how he entered the house of God ... and ate the bread of the Presence, which it was not lawful for him to eat nor for those who were with him, but only for the priests?
Luke 6:4 (ESV):
how he entered the house of God ... and took and ate the bread of the Presence, which ... is not lawful for any but the priests to eat, and also gave it to those ... with him?”
Currently, most scholars believe that Matthew and Luke edited Markian material separately, so both Luke and Matthew chose to remove the reference to the the high priest. The other edits are very minor and to clear up the grammar of Mark which, according to Dr. Ehrman, is poor. This one parallel strongly supports the Markian priority hypothesis as both later writers make obvious, yet different edits.
Why did both Matthew and Luke remove the high priest reference rather than substitute Ahimelech? One plausible reason is that 30 plus years of oral tradition insisted on Abiathar. It's one thing to omit a non-essential detail and it's quite another to alter it completely. The path of least resistance is to just drop the reference to the high priest altogether.
Another answer by jrdioko points to the idea that the phrase is midrashic or that it should be understood as a general marker of a time period. In these cases, Abiathar shouldn't be considered a mistake so much as an interpretation problem to be solved. Using the name of one priest rather than another might have been intended to carry a deeper significance. Unfortunately, that Luke and Matthew both drop the phrase indicates that, at the time they wrote, the reason for the substitution had already been lost. If it had been retained, the authors would have had reason to preserve the text or even elaborate on it.
Therefore, we must take seriously the possibility that Jesus misspoke, the Apostles faithfully recalled his error, and Mark faithfully recorded their oral tradition. The error was only noticed after Jesus' death and it was too late to ask him to clarify the teaching.
Finally, we must consider what it would mean that Jesus misspoke (assuming the above analysis is correct). One surprising result is that it might actually increase our estimation of the reliability of oral transmission of Jesus' teachings before Mark. Afterall, it is remarkably probable that someone would have noticed the discrepancy over the years and yet the high priest phrase was not dropped until after the oral tradition was made obsolete by a written account.
Second, it suggests that the written transmission of the New Testament was not irredeemably corrupted by later scribes. They had ample evidence that something was strange about Mark's account of this story and yet they continued to copy Mark's words faithfully.
As for what it means to the Christian faith, I find it comforting that Luke and Matthew recognized an error in one of their sources and yet did not lose faith. Any further would be verging off-topic, but I personally am not shaken by this passage in the least.