if are limited cases, can use Case Statement.
update yourTable set Last_Name = CASE When Name = 'James' then 'Jamerson'
                                 WHEN Name = 'Becky' then 'Beckerdude'..
                                 --then for each case
                                 END
edit
if you got the relation Name - NewLastName in another table, then create a script:
select
'When Name = '''+CAST(Name as Varchar(50)+''' Then '''+Cast(LastName as Varchar(50)+'''
from yourTableWithRelation
this will generate all When Name then LastName
then add  it to the update 
Update yourTable set Last_Name = CASE 
--Paste here the generated
END
edit2
another BEST way to update, if you got the relation in another table:
Update T set T.Last_Name = T2.LastName from YourTableToUpdate T inner join TableWithNewLastName T2 
on T1.Name = T2.Name