but my go to seafood restaurant in Boston is the Chart House on Long Wharf next to the Marriott there.

 
They may have cod, but such white flakey fish have unfortunately become a rarity even around here on account of over fishing.   Still, they will probably have something similar to it and they do a great job in preparation.   The place affords great views of the harbor and exudes old time Boston charm.   Be sure you get reservations if on a Saturday night.

I'm showing my knowledge of Boston history (or maybe that means I'm showing my age), but here goes.
Anthony's Pier 4 was a famous landmark seafood restaurant - closed in 2013.
http://en.wikipedia.org/wiki/Anthony%27s_Pier_4
.
The No Name Restaurant - closed in 2019
http://en.wikipedia.org/wiki/No_Name_Restaurant
.
The Union Oyster House - still open and operating! - is another Boston Landmark restaurant. You can see their gigantic "Union Oyster House" sign from far away.
http://en.wikipedia.org/wiki/Union_Oyster_House
.
Legal Sea Food - popular and famous Boston restaurant chain
http://en.wikipedia.org/wiki/Legal_Sea_Foods
Several locations in Boston and environs
.
There are many newer, smaller sea food places that are decades away from having a 100+ year history (they've only been around and successful for 20, 30 ... 50 years, so the verdict is still out on them   but I'm not up to date on them or their specialties, standout features, etc..  
.
http://www.themadmonkfish.com/ - Central Square, Cambridge.
Asian cuisine. Their specialty is their "Thelonious Monkfish." And they have live jazz.  
(Links for the jazz and humor impaired: http://en.wikipedia.org/wiki/Thelonious_Monk , https://en.wikipedia.org/wiki/Lophius )
.
You said you want "local "Boston/New England" cuisine" (Union Oyster House and Legal fit that) but I'll add that Boston has many great "ethnic" restaurants that serve fish on their menus -- Italian cuisine in the North End, Spanish restaurants, Dominican restaurants, Brazilian tapas houses, Japanese, Chinese, Thai, ...  
.
Sorry I can't provide a specific recommendation. Are you staying at a major hotel? Ask the Concierge for a neighborhood recommendation or up to date restaurant guide.

Recommendations for you

Eddie V's OR Ostra 😋

Here are a few that are actually still open 😀.  If you want a high-end experience with great food and service (very pricey but worth it) - Ostra in the Theater District.  If you want something funkier (especially if you like oysters) - B&G Oysters in the South End, Neptune Oyster in the North End, or Row 34 in the Seaport.  If you are in Back Bay try Saltie Girl which is a fun atmosphere and has a little of everything.  Ocean Prime and Lola 42 in the Seaport are more upscale and have extensive menus/great seafood.  There are a lot more to choose from but these are the ones I have been to recently.  Hope you enjoy your trip!

I second dire_wolf on Salty Girl: great food and atmosphere, but wasn't at all impressed with food at B&G Oysters, particularly considering high expectations we went there with. Brought a Korean from one of the two late Korean agencies there. The place was ok but the food was a let down. My date wasn't impressed too which was a bit of a bummer.  

My favorite seafood place in Boston is a very simple place, a diner really, called Belle Isle Seafood in Winthrop, a town on the other side of Logan Airport, about 20-30 minutes drive from the Downtown depending on traffic. If you're looking for a place to dress up this is not it. https://www.belleisleseafood.net/  

But if you're looking for a magnetic view, and great straight forward seafood right off the boat you'd be hard pressed to find anything better.  
The place is in the marina. There is dedicated pier right outside of the restaurant. The boats come in and the catch goes directly into the kitchen. All fresh and tasty.  

One of the best Clam Chowders in Boston imho. Get at least a cup because its very filling. But! You got to try their baked haddock dinner! A-la-la-la-la! You'd be licking your fingers long after leaving the place. You'd get few choices with dinner - their coleslaw is amazing! Crab cake dinner is also good but I'd go with haddock. Also, fresh and super delicious shrimp boat. I was a bit disappointed with scallops though.  

A side note: surprisingly good burger for those not into fishy/seafoody stuff.  

Cash only but there is an ATM on the premises. Also very economical considering the quality of food you'd be getting.  

Salty girl is great, imaginative, deliciously prepared food in nice semi-colonial atmosphere. Belle Isle Sea Food is as straight forward as it can get. No thrills - jeans, sweats, pajamas whatever. Just an amazing food with a view. Much better in the summer of course when you can sit outside and marvel, especially at dusk.

Welcome to Boston!  

I’d recommend the Union Oyster House. It is one of the older restaurants in America.  You will definitely find traditional New England style food. The place is rich in history.  JFK ate upstairs regularly.  
For dessert head over to the Omni Park House on School Street for the original Boston Cream Pie.

Want real New England dining? I make a mean baked cod, or baked stuff filet of sole, and a helluva great salmon steak.