Which Linux distribution would work well on an older HP Pavilion g6?

Try Lubuntu or Linux Mint Xfce from a live USB before installing. Lubuntu uses LXQt and is particularly light, while Mint Xfce is a little more comfortable for general desktop use. Both should work with Legacy BIOS, but test the display, Wi-Fi, suspend, audio, and networking before committing. The Pavilion g6 family has been used successfully with Linux, although exact hardware can vary between models.

Linux Mint with Xfce, MX Linux, Xubuntu, Ubuntu MATE, or Lubuntu would all be reasonable choices. Xfce and LXQt are especially suitable for 4 GB of RAM. The Ralink wireless adapter is generally supported by the Linux kernel, but boot a live USB first to confirm that Wi-Fi works properly on this particular laptop. Bluetooth will require a compatible USB adapter if the machine has no internal hardware.

Ubuntu LTS is another practical starting point because it has broad hardware support and lets you install different desktop environments without reinstalling the operating system. You can start with Xfce, MATE, or LXQt if the default desktop feels slow. Linux is a sensible way to keep this laptop useful after Windows 10 support ends, especially if you pair it with an SSD.

Four gigabytes is usable for a lightweight desktop and ordinary tasks, though modern web browsing can still consume most of it. If the laptop supports it, increasing the memory to 8 GB or more would help, but an SSD should be the first upgrade. Enabling zram or configuring a modest swap file can also make the system behave better when memory is tight.

The biggest performance problem is likely the mechanical hard drive rather than Linux itself. Replacing it with even a small SATA SSD will make booting and launching applications dramatically faster. You can keep the old drive for storage if the laptop supports installing both drives, or use an external enclosure.