Travelling (as a family) with children with different surnames
There is nothing wrong with travelling with a child with a different surname to yourself. In fact there are many countries in the world (e.g. Italy) where it would be unusual for a mother to have the same surname as her children, and where it is perfectly legal for her to travel (alone) with them.
It can still happen though that you be questioned — at a border control in the U.K. or overseas — about whether your child is indeed yours.
To avoid this problem we would strongly recommend that, at the very least, you enter both parents’ (or guardians’) names and contact details in the “emergency contacts” section at the back of your child’s passport. This costs nothing, and it also prevents anyone else (i.e. a would-be kidnapper) from doing so.
If you are travelling with someone else’s child (and the child’s parents / guardians are not travelling with you), we recommend you carry a signed letter from one or both parents / guardians, confirming that they have given you permission to travel with their child. The letter should include the contact details of every parent / guardian with parental responsibility.
Although your child’s (full) birth certificate does prove who their parents are, it is not a legal requirement to bring it with you when you travel with your child. Bringing it with you is a hassle, and it means you might lose or damage it. It is, nevertheless, the most effective way of proving that your child is yours.
As a justification for changing a child’s surname (to your own)
Bear in mind that — on its own — wanting everyone in your family to have the same surname is not considered (by the courts) to be a justification for changing their name. That is not to say that there aren’t benefits of a child having the same surname as everyone else in the family (including any other children). But the mere fact of a child’s surname being different, and the day-to-day inconvenience of that being the case, is not a sufficient justification for a change of surname.
Neither is it (nowadays) considered socially unacceptable, generally speaking, for a child not to bear the surname of one or more of its parents (or other children in the family). There are many families where parents may have different surnames to each other and/or their children, and children themselves may not bear the same surname as each other. Notwithstanding, there coud be specific cultural and/or religious reasons, specific to a child, that override this general rule.