Mac Book 2010 won't recognize My Passport for Mac

Welcome!

It worked relatively fine for many years,

I'm concerned about the phrase, "worked relatively fine." I demand 100% from anything backing up my data. Plus "many years' is a red flag for that type of drive.

Inexpensive, bus-powered external drive—including Passports— have a predilection for dying before the computers they serve. They also tend to demand more power as they age, more than the computer's usb ports can provide.

Some options for the low-power issue if present:

• Most such drives ship with Windows formatting. If you do not need to share the drive with a Windows computer, you can use Disk Utility to erase and reformat the drive as Macintosh HFS Extended. Some have found that helps.
• You can buy a POWERED USB hub—comes with an independent power supply— and put it between the computer and the Passport.
• If you have enough open USB ports (maybe not on a 13-inch Macbook Pro), you can try a "Y" USB cable that connects to TWO ports on the computer to give more power to the Passport. Example here: Y-USB cable

As you see, two of those options require spending money on a drive that could have died of old age.

If you need to replace this drive, most of the senior contributors here recommend the Mercury models that OWC sells; I concur. I have five OWC backup drives serving our three "mission-critical" Macs. They have never failed or given even a whiff of trouble.

If you wish to check out what OWC offers:

OWC desktop drives

OWC portable drives

Those are from the US site. OWC also has world partners.

Disclosure: I have no relation with OWC other than as a long-term customer.

Does it appear in Disk Utility: Disk Utility User Guide for Mac - Apple Support

Describe those "issues". When a hard disk drive becomes unreliable, you can try reformatting it, but when those failures become intolerably frequent it's time to discard it.