Really not enough information to go on here. Color, whether the wood was quartersawn or not, what's happened in the last 104 years can all effect.

Given the age, wood, and the fact that it's a picture rail, I'm assuming it's a Mission / Craftsman finish. See the link for some ideas and possible colors.

General Finishes' Java gel stain is also a good choice for a one-can mission finish.

Here is a link that might be useful: Mission Finish

Try aniline dye.
You can buy the base colors and mix it to get just about whatever you want.
Mosers is one vendor.
Woodcraft Supply is one source.

hate to say it man- but there is no magic bullet here. Im a professional furniture maker and all i can say is you have to experiment. minwax makes a stain designed to make white oak look like red oak- but this will not work in your case because the the red oak you already have has been stained. are you certain that it is red oak? in older homes white oak is much more prevalant than red oak.

just remember- as you experiment - write down your methods so that you can repeat what you do when you find the right color.