I believe there are some unofficial backports of Firefox and Chrome (or rather Chromium) web browsers which will run on your laptop, but I am not personally familiar with them. IIRC there is a critical unpatched vulnerability with OSX 10.7.x Lion, but I don't recall the details, but it is serious enough that we are not allowed to use it in our organization.
If you want an up to date OS on this laptop, you could try installing Linux on it (Ubuntu or Linux Mint). Both of these Linux distributions should be easy to install and run on the MacBook from what I've been told. They have access to the latest versions of the common web browsers (Firefox, Chrome, Vivaldi, and others). It also includes LibreOffice by default which tries to be compatible with MS Office since MS Office doesn't run natively on Linux, although some 2010 and older versions of Office can be made to run using WINE (available with Linux) or Codeweavers' CrossOver app (paid app).
Unlike Windows and macOS, Linux has access to many different graphical Desktop Environments. Desktops Environments are just the graphical interface you see and interact with. The underlying core OS is the same with each Desktop Environment. The Desktop Environments which I've found that will run with limited memory and minimal CPU are Mate, KDE, LXDE, and XFCE with KDE & LXDE modeled a bit like MS Windows. While I don't use either distribution, the version of Linux I do use has these Desktop Environments using only about 300MB at login. Do not attempt to run the standard plain Ubuntu with the Gnome Desktop as it requires a high end system to run. There are other Desktop options as well. Linux is free to download and use, but will require you to learn the OS as it is different than OSX. Linux is not for everyone, but since it is free it is easy to try it out especially since you know how to reinstall Lion if you want to go back.
You can create a bootable Linux installer from the downloaded .iso files using Etcher (Mac/Windows/Linux). Etcher requires OSX 10.9+ to run, or you can use the Terminal "dd" command to burn the installer .iso to the USB drive. Option Boot the Linux USB installer and select the orange icon labeled "EFI".
Ubuntu Mate -- Ubuntu with Mate Desktop
Kubuntu -- Ubuntu with KDE Desktop (tons of customization options)
Lubuntu -- Ubuntu with LXDE Desktop (very minimalistic and lightweight)
Xubuntu -- Ubuntu with XFCE Desktop (fairly minimalistic and lightweight)
Linux Mint -- Linux Mint (based on Ubuntu, with Mate and XFCE desktops currently available)