I have strong doubts it is possible to install Debian or Ubuntu in your DVR.
Ignoring the fact most often some consumer-grade processor/devices have a shoddy job in the kernel department with some hacked/proprietary device drivers, busybox is usually used where there are severe limitation of resources. You might have lucky and have a more supported device, however it is not the norm.
I have an embedded Linux at home, with 128MB of RAM, and 16MB of disk; yours seem to be aprox 96MB of RAM + 128MB of disk. With this kind of resources there is not much leeway to install a normal Debian or Ubuntu distribution without significant work.
Even then, I have to point out that seeing a Linux prompt somewhere does not means vanilla kernel supports it, or that there are open source drivers for it. (nor does it means there are maintained closed drivers for it).
The string stm24_V2.1-SDK7105_7105-STSDK in your
uname command led me to find out it seems a Chinese distro for embedded devices, STLinux, based in the linux kernel 2.x ; again the same comments apply here, we cannot say for sure without documentation until which extent it has been hacked. STLinux seems to depend on rpm packages, and appears to still have security updates.
The STLinux distribution and development environment provides
everything required to build Linux based systems for
STMicroelectronics products which are based around the ARM Cortex A9,
ST40 or ST200 CPUs.
The STLinux page also will provide you hints how to deal with your board. Also according to them, they are not using grub, but u-boot.
The U-Boot utility is a multi-platform, open-source, universal
boot-loader with comprehensive support for loading and managing boot
images, such as the Linux kernel.
You can try to upgrade it at your own risk. I would say the probability of bricking the device is high.
As for interacting with embedded consumer devices, often there are JTAG ports under the hood (i.e. opening the box), which provide RS-232 access to the environment, often before the kernel booting. If so, they can be used with a cable similar to the PL2303 PL2303HX USB to UART TTL Cable Module 4p 4 Pin RS232 - the cable itself costs around 2USD in aliexpress.
Your mileage may vary, I would google around the maker and model of your box.