I haven't seen any discussion of electrifying the B&M, though that's certainly not out of the question. The New Haven had Back Bay and South Station built for electrified suburban service, and that tunnel would have certainly required it.
The New Haven never actually did much towards electrification in Boston, actually - they had the Nantasket Beach line under trolley operation from 1895 to 1932, with some of the South Shore line with brief electric service between Braintree and Cohasset from about 1898 to 1906.
Lynn did see electric service - the BRB&L was successfully electrified in 1928, but the poor timing of the depression eventually killed it.
The NYC actually strongly considered electrifying the Worcester main as far as Framingham, as well as the Highland Branch, in 1911. They decided that the cost wasn't worth the savings. The Newton Lower Falls branch was electrified as a trolley from 1904 to 1930, though.