For compact, they will not want a DSLR camera. While I love mine, it's by no means compact. I have a camera bag that's about the size of a large woman's purse (a large purse, not a purse owned by a large woman - felt the need to clarify that). In it, I have the camera body with standard lens and the zoom lens is on stand by. At cheer competitions, I toggle between the two depending on where I'm set up. The speed is near video quality and I can take HUGE pictures (can get the entire stage in the view finder and then go back and zoom into individuals later).
For portability, I'd look for a point-and shoot camper like the Canon Powershot line-up. A lot of them can be stuffed in a pocket, but still have very nice zoom capabilities. A lot depends on the user and what they like. There are some point and shoot cameras that are even water proof. I don't know of one DSLR that is not without dropping thousands on a pro setup.
If you think that photography might become a hobby/ potential work interest, I'd go right for a DSLR and not look back. If they are just going to use it for vacation photos, selfies, pictures of the family dog, then go point and shoot.