A TV with a Bluetooth remote will be useless for controlling the cable box. You will still have to use the cable box remote to change channels. Yes, there are some TV's that attempt to control your cable box with an IR blaster but these systems are clunky and confusing and will most likely not be intuitive for your mother. That's not to mention the fact that you are unlikely to find such a feature on a small TV.
Your best option would be a universal remote with RF control. These systems connect to a basestation that then repeats the commands to your individual components so you do not have to point the remote at anything. Your cheapest option for this would be something from Harmony which you can install and setup yourself. The problem with this (and any professionally installed system) is that old Phillips CRT likely does not have discrete power on/off commands. This is important in a universal control system because it makes it much more reliable. When the remote can send a "power off" command that ONLY turns the component off or a "power on" command that ONLY turns a component on then the macros will be much more consistent.
When there are no discrete commands and the remote must rely on "toggle" commands then things tend to get out of sync and devices will turn off when they're supposed to turn on, vise versa, etc. This can become very frustrating. To add to that, many small TV's from even major manufacturers for whatever reason don't respond to the same discrete codes that other larger models from the same manufacturer do so even a new small TV may not solve the problem.
A professionally installed remote system like URC, ProControl, or RTI would be a good option although they are significantly more expensive and must be installed and serviced by a dealer. The problem with discrete commands would still exist but could be worked around using sensors that tell the control system when a component is on or off. All this would be a significant expense to put toward an old CRT or a very small 24" LCD.
