One time we decided to try the oldest restaurant in the United States, Ye Olde Union Oyster House. The food was good, yet the price (at least for us) was pretty high. There are many better seafood restaurants in Boston, although none can beat the historical attraction of the Oyster House.
This restaurant makes great Clam Chowder (CHOWDAH to locals), arguably the best in the city. However, it's located right on the Freedom Trail near Faneuil Hall Marketplace, and that means that it's constantly crammed with tourists. The gift shop in the front and various random historical data on the menu can give it a kitschy feel, but it does have historic significance, having existed as an establishment since 1836. For a more authentic Boston dining experience closeby, try Durgin Park in the upstairs of the North Market Building of Quincy (Faneuil Hall) Market.
Fun Alternatives: If you want to experience the city's best chowdah, come to Boston during Chowderfest (July 4 week at City Hall Plaza) for a sampling of chowdah from many local restaurants.