Upgrading from 10.9.5 to Big Sur

The Age of this computer will Govern what newer version of macOS it will Qualify to install. Use About this Mac from the Apple Icon on Desktop - the required Information will be there.

With the above information in-hand - follow this How to get old versions of macOS and verify what version this computer Qualifies to install.  For Best results use Safari to commence the download as Others may not work.

Would strongly suggest upGrading to High Sierra FIRST as the OS installed now to too Old to make the leap directly to Big Sur.

Installing High Sierra may better position the computer with a possible Internal Drive conversion from the Old HFS Journaled / GUID to the newer APFS / GUID format which is Required to install Mojave10.14 , Catalina, 10.15 and Big Sur 11

It may also bring down EFI BOOT ROM updates to allow the computer to function and understand the newer Drive Format.

Safari will open a redirect and open the Apple Store where the download can commence

Tips Big Sir Upgrade Released Nov 2020

Make Time Machine Backup  is very useful and can be used to Revert to Previous Working macOS

Notation - Mojave is the Last version to support 32 bit applications / extensions and Drivers. Catalina, Big Sur & Monterey supports only 64 bit and there is no workaround.

Suggest using Go64 to find 32 bit application before attempting the upGrade. Some or many of the Application already installed may required update or upGrades to 64 Bit Versions.

Are there any AntiVirus, Disk Cleaner, Optimizers, Un-installers, etc installed which should be removed as per Developers Instructions. They are useless, unneeded, cause havoc and interfere with the normal  operation of the OS and may even Corrupt the OS requiring a Reinstallation

I have seen the issue you describe posted otherwhere, and there is (most likely) a cure. First things first - let's check something. 1) Make sure the Mac is in Finder (look next to the Apple icon in the extreme upper left of the screen. If you see something else, click once on the desktop. Once in Finder, from the upper menu, choose Go, Applications. Once in the Applications folder - look for Install MacOS Big Sur. If it's there, check it's size - left click once, right click and choose Get Info. See if it's 12.5 GB. This will verify that you have the full installer. From your screenshot, I am pretty sure you actually do. But check.

Let me know. I should warn you that MacOS Big Sur will not run any 32-bit apps, nor will a printer or scanner with a 32-bit driver work. I helped a friend install Big Sur this morning, and getting a driver for their printer was a problem for a while.

Glad to hear it worked out. The most commonly recommended workaround for your issue I've seen is to first install High Sierra, as per P. Phillip's reply. I hope that you have smooth sailing from here. Hopefully, if your wife had old third party software she was using, they are still being maintained, in order that she can obtain 64-bit replacements. Not a problem at all with something like Firefox, but old/obscure software is problematic.