This looks like the right record.I think he was raised in NH and moved to Boston because he was a businessman.
GI Individual Record FamilySearch™ International Genealogical Index v5.0 North America Search Results | Download | Pedigree
WILLIAM A ALDRICH Male
Event(s): Birth: 27 MAR 1824 Westmoreland Twp, Cheshire, New Hampshire
Christening: Death: Burial:
Parents: Father: WILLIAM ALDRICH Family Mother: LYDIA ARNOLD
Messages: Extraction of birth information from a statewide index for New Hampshire. The actual town records may contain more details. Use the Family History Library Catalog to find the call numbers for town records.