I disagree on the .243 being marginal for deer. This might be partly due to the deer here in Texas generally being smaller. Keeping in mind that many are having good results with .22 centerfires,makes the .243 a MAGNUM. My grandson started at age 9 and his first few deer were with .22 centerfires. He moved up to the .243 and has used it the past several years and has taken atleast one deer every year. I too use a .243 and have found it adequate. I am going to give him a .308 as I happen to have a Win 88 that I think he would like. Moving him on up to .308 is in no way saying the .243 has not been ample. More important than using a cannon is shot placement. If a person whether 9 years old or older is shooting something that is smacking them they probably won't shoot so good. There will be those that say oh no don't use a .243 you must use atleast a -----------fill in the blank. Use one of those filled into the blank and hit em wrong and you have problem. The only conclusion I have reached is you don't know what will happen until you have pulled the trigger regardless of what you use. You asked about other options--the 6mmRem is a little ahead of the .243--doubt that deer have read about it and wouldn't know the difference. The .257Roberts and its dad the 7x57 are both fine rounds--maybe too much recoil for a 9 year old.

I totally disagree that the .243 is "marginal" for whitetails. I say this with all due respect and don't mean to offend anyone. But, I believe that more deer are killed with .243's using the wrong bullets. With the proper bullet, the .243 is devestating on deer. One of the top choices is the Sierra 85 grain GameKing, driven at .243 velocities, it will knock down a deer with authority. Now, I'm not going to set here and say that the .243 is the ideal caliber for deer, that's generally believed not to be true. It is quite capable of making clean kills without giving up much, if anything.
An interesting side note is that Sierra used to list the 85 gr GameKing as a varmint bullet, until so many people deluged them with mail about how well it worked on deer, so they changed it's designation to GameKing.
Although I haven't had a chance to use them yet, I've been experimenting with Speer 100 grain Hot Cor's and will possible fling a few at deer this year.
In all honesty, I have other favorites for deer, mainly the 7mm-08 and 270W, but I've killed several with the .243 and have learned over the years, which bullets work.

.250 Savage (Short action)
.257 Roberts (intermediate length though)
7x57 Mauser (same problem)
.260 Rem
7-08 Rem

Oh the choices....If you are not confident in the .243 and want the most from a true short action though, the .260 and 7-08 are your best chouces...IMHO of course.