Okay, I know it is not enough time but my daughter will have one day to spend in Boston next month. What would you suggest - she will be relying on public transportation and
is not familiar with the city.

Having been to Boston many times I would do the following if I had a short amount of time. Keep in mind Boston is a great walking city so the "T"(public transportation) may not be needed.
1. Walk Newbury street
2. Check out the view from the Hancock
or Pru.
3. Eat some seafood at Legals or any
other seafood joint. Also their is
good Italina food in the North End.
4. Walk around the public garden or
Commons.
5. Take the T to Harvard sq. Although
it's not what it used to be.
6. Stay away from Cheers.
7. Go to a sporting event if possible
8. Enjoy

CJ's advice is good, although I would add a walk from the Public Garden up to the State House and around Beacon Hill via Louisburg Sq. down to Charles Street -- perhaps instead of Newbury St., which is $$ and upscale shops (unless that's what she'd like).

Emphatic ditto to CJ re: avoidng Cheers.

Walking and view from Hancock or Pru depends on good weather. Legal Seafoods is at the Pru (and other places) and is better than many of the obvious tourist seafood places on the waterfront.

If weather's less good, take T to Harvard Sq. and hang out in the Coop with a cup of tea or coffee, a scone, and watch the people and/or check out the books. If she likes art at all (including modern) and/or would like to see a good museum shop, the Museum of Fine Arts on the Huntington Ave. line (Green, but make sure of correct line) is also a great indoor visit.

I'd suggest going to the Boston Commons and walking the Freedom Trail (free) to Faniel Hall. If I remember it's approx. 2 miles but you'll get to view John Hancock, Paul Revere,etc. grave site, one othe oldest churches in U.S., & end up at Faniel Hall. Very easy & pleasant walk.

Ditto on walking the Freedom Trail, since this IS a first visit, and one day at that. Boston's MAIN attraction is its history, and to skip the FT is to miss the whole point of visiting there.

The ONLY thing I might recommend instead of the FT is the Museum of Fine Arts, and only for someone more interested in art than history. If that's the case, see the Isabel Stewart Gardner Museum as well. These two will make a full day, so don't plan on anything else.

It's up to "san's" daughter, but there's a lot more to Boston than colonial/revolutionary history -- things have happened since 1799. It's certainly worth doing the Freedom Trail at some point, but if you only have one day in Boston, I'm not sure it's the best way to get the flavor of the city.

Of course, we don't know what "san's" daughter likes or even how old she is, so it would be helpful to know. If she's college age, the biggest "news" about Boston is that it's full of colleges and perhaps she'd prefer the areas around Harvard, MIT, BU or . . . If she likes music, perhaps she should make a point of hearing the Boston Symphony. People have already mentioned the MFA for art, but I'm not sure I'd go to the Gardner if I only had one day in the city. Sports = Bruins (on a good day, maybe) in Oct.

I like CJ's suggestion that walking is the best way to do it, and the itineraries CJ and Pris suggest are good.

About how long does it take to walk from the Public Gardens and the Freedom Trail to Fanueil Hall? Are Are the PG and the FT along the same path as each other?

The Freedom Trail starts on Boston Common, which is one block East of the Public Garden. I believe the Freedom Trail starts near the Park Street T station on the Common. As far as walking time is concerned, it depends. If you want to stop and see the historical sites on the way it will take longer. If you just decide to walk without stopping, it's only about a 10 minute walk. Have fun!