What would you do if one member is sitting silently doodling on a piece of paper?

I suppose you mean a member of your family?

It depends on the situation. Is she/he enjoying the doodling or is she/he doing it to mope, or does she/he have a habit to disconnect from others?

If she/he is enjoying the time doodling, you could go there and see what she/he is doing and comment on the thing. Or you could just leave him/her in peace to do it.

If he/she is moping. Let her/him to cool off first. Half an hour maybe. Then go to him/her and ask what is wrong and let her/him pour it on you.

BUT if you are the reason why she/he is moping, you should apologize the situation (sorry that her feelings were hurt) and explain why she/he was denied the whateveritwashe

/she wanted. Especially if you are a parent, you do not need to apologise though, but an explanation is in order. If that does not help, let him/her be and cool off. He/she will come around at her/his own time.

But if she/he is shy or otherwise disconnected from others and has a habit to isolate her/himself, you should ask him/her why. If he/she is too young or cannot explain it, you should observe him/her a longer time and see if this is a passig phase or something that needs further investigation.

If she/he can make normal human contacts and behaves normally otherwise there is no reason for consern. But if she/he has difficulties communicating with other people, acts passively to everything or is self-destructive, you should seek for professional help.