Don't know if they're open on the 1st, but two Boston board favorites in Somerville are Sound Bites and Ball Square Cafe, both in Ball Square. BSC is new, having moved into the former SB space and hiring its former chef; SB is the old reliable. Both have more or less the same MO, serving better-than-average takes on the usual egg 'n' griddle fare.
The fresh fruit is what makes both eateries pass my could-I-easily-have-cooked-a-better-breakfast-at-home test; order anything with fresh fruit and you'll get a surreally large mound of assorted fresh fruit piled on your dish. BSC is my preference, as the owner is actually nice, and not the caricature-like "Breakfast Nazi" you'll find at SB; I also prefer BSC's coffee.
In Boston, two good hotel restaurants that would surely be open, serving a good, Europhilic brunch are Eastern Standard and Beacon Hill Bistro.
Beacon Hill Bistro
25 Charles St, Boston, MA 02114
704 Broadway, Somerville, MA 02144
Ball Square Cafe
708 Broadway, Somerville, MA 02144
We're looking too and have called around to a couple places. Seems like a number of restaurants are closed tomorrow, kind of a bummer. If you don't mind a trip into Kendall Sq., the Blue Room (on Zagat; www.theblueroom.net) is serving brunch from 12-4. Their brunch is pricey, but fantastic. Great way to ring in the new year.