I understand that some names like Cohen for instance, are special in some way. But what I don't understand is, if you can only be Jewish through your mother, how can your last name ever mean anything?

Answer

Jewish identity passed on through the mother has been universally accepted by Jews for 3,000 years, and was decided by God. The rabbis and the people of Israel simply passed on the information from generation to generation. This is recorded in the Five Books of Moses in Deut. 7:3-4. The Talmud (Kiddushin 68b) explains how this law is evident from those passages.

In another sense, the father passes on lineage as well, concerning which tribe the child belongs to. This will determine whether the child is a Cohen, Levite, or Yisrael. See Numbers 1:20-46 where it explicitly categorizes the Jewish people by their "father's house."

It should be noted that just because someone's last name is "Cohen" doesn't mean that he has the status of a Cohen. To be considered a Cohen, one must have an unbroken tradition, as well as other factors too numerous to mention here. Nevertheless, it does turn out that many people who have the name Cohen also have the status as Cohen.

From the fact that the religion of the child goes by the mother, while the tribal affiliation goes by the father, we see that both the father and mother must take active roles with the child. The mother is entrusted with the awesome duty of instilling in the child faith in God, observance of mitzvot, and Jewish pride. By way of metaphor, we see that the mother gives the baby food and love that brings out it's internal potential. This is in contrast to the external qualities, represented by tribal affiliation that is the father's duty.