I haven't changed my name and the kids have their dad's name. It wouldn't have worked to have my surname as a middle name (eg: Fitzpatrick Fitzsimmons).
My kids are now 14 and 12.
When I travel alone with the kids I take a photocopies of their birth certificate, their father's passport and a signed letter from him granting permission for us to travel without him.
In some countries travelling solo with a child is what triggers a check, rather than what names you do or don't have so it's a good idea to have these anyway (and in the case of a parent that has sadly passed away, a copy of the death certificate).
I've only ever been asked 3 times, twice by British Airways(!!) and once by UK border control on entering the UK.
I personally think it's ridiculous to rely on matching surnames as a way of ensuring kids are not travelling illegally. I have a friend who has gone through a nasty custody battle - his wife took his surname but is a risk for absconding with the kids whereas me and my different surname are no such risk. It would be sensible to have a standard procedure/set of checks/documents for all kids not travelling with 2 parents.
There is a German website (I think their version of the AA) that offers standard forms for this - although the website is in German language, the forms are multi lingual (including English):
ADAC forms
You could use this - or copy the English terms if you preferred. The middle pdf is the form for a child travelling with just their mother.
Some countries are extraordinarily strict about travelling with children - eg: South Africa requires you to present the LONG form version of a child's birth certificate, regardless of both parents travelling and your matching surnames - if you have no such thing, you can't travel (and from the experience of a friend, your Xmas plans will be ruined as a result!)
Long story short: changing your name on marriage actually doesn't insure against issues re: crossing borders with children.