demo.rb

objects = [MyObject.new(name: "John Smith"), MyObject.new(name: "Joan Smith"), MyObject.new(name: "Jane Doe")]

objects.select {|o| o.name["Smith"]}.each do |o|
  o.name.gsub!("Smith", "Jones")
  o.save
end

The gsub! is important, without the ! the gsub will return a new string, rather than change it in place. So these are equivalent:

o.name.gsub!("Smith", "Jones")
o.name = o.name.gsub("Smith", "Jones")

Also, I'm assuming you're not using ActiveRecord... If you are then you'd do something like this:

MyObject.where("name like ?", "%Smith%").each do |o|
  #  ...
end