The Doctor never explicitly states that he is going to take Amelia with him but she is certainly left with that impression. After he departs Amelia rushes to fill a suitcase, believing that he’ll return within five minutes, just as he promised. Sadly it would be years later before she saw the Doctor again.
Which raises the question of if the Doctor would have taken a young child as his new companion. We’ve seen that the Doctor can be overconfident in his skills at piloting the TARDIS and so might not be concerned with Amelia being missed, believing he could return her before she is missed.
There is the issue of whether he would put a child into danger, as so often happens to his companions in his adventures. This can be explained if we take into account that we only see the exciting trips through space and time. These could be much more infrequent that we are led to believe, meaning that it could safe to take someone on a few trips before returning them home.
It must also be remembered that the Doctor is both an alien and hundreds of years old. The difference between a child and someone in their twenties is insignificant to someone as old and long lived as the Doctor.
Having a child as a companion only strengthens the fairy tale feel of Doctor Who. The companion often serves as our point of view character and a young companion couldn’t help but add a sense of wonder to every adventure.
The change in the dynamic between the Doctor and his companion would also create adventures with a very different feel. The Peter Cushing film ‘Dr Who and the Daleks’ illustrates this, with Susan being portrayed as a little girl and Dr Who acting as her grand fatherly guardian.
While lacking physical strength, competency or skills a child does have advantages. Chief among these is that they aren’t viewed as a threat. This can spare them from violence and put them in advantageous position due to be being under-estimated.
In any conflict the attacker would have to have a complete lack of morality to harm a child. Those motivated by logic or a concern with resources will also evaluate how worth while it is to harm a child that presents little danger to them.
Unless the child endangers their plan even a Dalek might not exterminate a fleeing child. Not when there are more important things to be concerned with. Similarly the Cybermen wouldn’t see a child as a worthwhile subject for conversion and likely ignore them.
At most the child will be captured, either to keep them out of trouble or to extract information from. The enemy could use a child captive to blackmail the PCs. All of which gives the child time to collect information and engineer their escape.
Wild animals present more of a hazard but within such stories such beasts are strangely calmed by the presence of children. This can lead to the animal being befriended, especially if the child were to feed them or remove a thorn from their paw, giving the child a valuable asset.
When all hope is lost and the child is about to be badly hurt, that can be the moment that a friendly NPC swoops in a saves them. This can be a native who can act as a guide to the area and reluctantly protect the child until they can be returned to their guardian.
Children can also do many things that an adult can’t. For example notice how in ‘The Daleks’ it is Susan who is tasked by the Daleks to retrieve the items from the TARDIS. They believe that she poses no danger and will do as they command.
A child will be ignored in most public places, allowing them to observe valuable information and listen in on private conversations. Even if the child is out of place in the location, acknowledging them would require the NPC to accept some responsibility for them. Many would rather turn a blind eye than deal with that.
Security guards might let a child walk into secure areas if given a valid excuse. They might claim they are visiting a parent or delivering a requested item or supplies. What security guard would suspect that the child is a spy capable of sabotage or working to free a prisoner?
A child’s small size can allow them access to areas that adults can’t get to. While anything too high up is out of bounds a child is capable of crawling through narrow gaps or hiding in small spaces.
Although a child doesn’t bring a great deal of knowledge their ability to contribute to the thought process can’t be under-estimated. Unfettered by ingrained thinking process they can come up with new ideas that others can’t. Their idea might not be workable but it can provide the inspiration an adult needs to solve a problem.
They can influence NPCs, questioning their actions and suggesting courses of actions. A character might be more easily persuaded by the innocent request of a child than they would be from an adult. If all else fails the child could use guilt to gain their compliance.
In an on-going campaign a player might create a character who is a child. This can set them apart from the others but can led to them being a liability. If everyone else is an adult the child PC might be pushed into the background while the grown-ups deal with problems.
The games master must ensure that their adventures give the child PC plenty of opportunities to contribute to the plot. They must be given something to do other than being put in danger or kidnapped.
An easier option is to have a child-centric campaign, with all the players taking the role of children and the Time Lord acting as an NPC. While a group of children travelling with the Doctor, or similar Time Lord, is possible you may wish to ground the adventures in a particular time and place.
This can take the format of ‘The Sarah Jane Adventures’ with a younger cast of characters. Here you contrast their everyday life (parents, older siblings, school) with the magical adventure offered by a time machine.
The Time Lord could visit them occasionally, whisking them away from their troubles. A games master might wish to have the adventure parallel their problems at home, almost as if the Time Lord is teaching them life lessons.
For example a PC who has been bickering with an older sibling could be taken to a world consumed in a civil war between two brothers. Someone who can’t relate to their parents could be taken back to meet them as children.
The Time Lord could take the PC into the past to help with their history home work or into the depth of space to complete a science project. This can help the PCs better appreciate the subjects they are studying and learn why that knowledge is important.
Only the child PCs should be aware of these adventures. The Time Lord whisks them away when no adults are around and returns them before they are missed. The PCs know that if their parents ever found out they’d either be prevented from going or thought to be crazy.
The Time Lord could be a neighbour. Perhaps he retired to Earth or was exiled. While everyone else thinks he is an eccentric old man the PCs know about his time machine at the bottom of his garden that is disguised as a garden shed.
One of the PCs might be related to the Time Lord. This could be their grand parent or their mother or father. Did they stumble upon the alien nature of their relative or were they let in on the secret?
If it is a blood relative does that mean that the PC is an alien as well? What issues would face an Earth-borne Time Lord? Would they want to visit their home world or be troubled by their inhuman nature?
Of course you can have adventures where their two worlds collide. This can lead to aliens intruding into their lives, such as taking the form of school teachers or a UFO landing in their local park.
Their adventures can make them ideally suited to dealing with these problems. This is especially true if they’ve encountered the alien race in a previous story. Can they deal with it on their own or will they have to find a way to contact the Time Lord character?
If you wish to bring this idea into an on-going campaign you could reveal that an adult PC has repressed their memories of these early adventures. This allows you to run flashback adventures, showing the characters as children.