Use the script posted by Thomas Lee in the comments of this link to get some inspiration on how to change these connection strings. Or recreate the maintenance plans.

Use the script (reproduced below) posted by NancySon in the comments of How to: Rename a Computer that Hosts a Stand-Alone Instance of SQL Server to get some inspiration on how to change these connection strings. Or recreate the maintenance plans.

Maintenance plans don't get their connections changed to the new server name and so they may break. After a rename you may find that you cannot delete or rename the existing maintenance plans, so either delete them before renaming the server and recreate them afterwards or run the following script to fix them:

   use msdb

   DECLARE @oldservername as varchar(max)

    SET @oldservername='<server name>\<instance name>'

   -- set the new server name to the current server name

   declare @newservername as varchar(max)

    set @newservername=@@servername

   declare @xml as varchar(max)

    declare @packagedata as varbinary(max)
    -- get all the plans that have the old server name in their connection string
    DECLARE PlansToFix Cursor
    FOR
    SELECT    id
    FROM         sysssispackages
    WHERE     (CAST(CAST(packagedata AS varbinary(MAX)) AS varchar(MAX)) LIKE '%server=''' + @oldservername + '%')

   OPEN PlansToFix

   declare @planid uniqueidentifier

    fetch next from PlansToFix into @planid

   while (@@fetch_status<>-1)  -- for each plan

   begin

    if (@@fetch_status<>-2)
    begin
    select @xml=cast(cast(packagedata as varbinary(max)) as varchar(max)) from sysssispackages where id= @planid  -- get the plan's xml converted to an xml string

   declare @planname varchar(max)

    select @planname=[name] from  sysssispackages where id= @planid  -- get the plan name
    print 'Changing ' + @planname + ' server from ' + @oldservername + ' to ' + @newservername  -- print out what change is happening

   set @xml=replace(@xml,'server=''' + @oldservername + '''','server=''' + @newservername +'''')  -- replace the old server name with the new server name in the connection string

    select @packagedata=cast(@xml as varbinary(max))  -- convert the xml back to binary
    UPDATE    sysssispackages SET packagedata = @packagedata WHERE (id= @planid)  -- update the plan

   end

    fetch next from PlansToFix into @planid  -- get the next plan

   end

   close PlansToFix

    deallocate PlansToFix

  ----- This will also handle the packages that have a tag such as 

    ----- <DTS:Property DTS:Name="ConnectionString">Data Source=servername;Integrated Security=SSPI;Connect Timeout=30;</DTS:Property>


  

   DECLARE @oldservername as varchar(max)

    SET @oldservername='<server name>\<instance name>'-- set the new server name to the current server name
    declare @newservername as varchar(max)
    set @newservername = @@servername
    declare @xml as varchar(max)
    declare @packagedata as varbinary(max)-- get all the plans that have the old server name in their connection string
    DECLARE PlansToFix Cursor FOR 
    SELECT id
    FROM sysssispackages
    WHERE (CAST(CAST(packagedata AS varbinary(MAX)) AS varchar(MAX)) LIKE '%Data Source=' + @oldservername + '%')

   OPEN PlansToFix

    declare @planid uniqueidentifier
    fetch next from PlansToFix into @planid 
    while (@@fetch_status<>-1) -- for each plan 
    begin
    if (@@fetch_status<>-2)
    begin
    select @xml=cast(cast(packagedata as varbinary(max)) as varchar(max)) 
    from sysssispackages where id= @planid -- get the plan's xml converted to an xml string
    declare @planname varchar(max)select @planname=[name] from sysssispackages where id= @planid -- get the plan name
    print 'Changing ' + @planname + ' server from ' + @oldservername + ' to ' + @newservername -- print out what change is happening
    set @xml=replace(@xml,'Data Source=' + @oldservername,'Data Source=' + @newservername) -- replace the old server name with the new server name in the connection string
    select @packagedata=cast(@xml as varbinary(max)) -- convert the xml back to binary
    UPDATE sysssispackages SET packagedata = @packagedata WHERE (id= @planid) -- update the plan
    end
    fetch next from PlansToFix into @planid -- get the next plan
    end
    close PlansToFix
    deallocate PlansToFix