From the 50's, it may be better to replace it as you indicated. But, replacing the cartridge should return it to it's as new function, but not appearance. That can be challenging, depending on the type as HJ mentioned.
I like Grohe and Delta. I think the Delta may be more flexible, and is usually easier to get parts for. At a plumbing supply store, the Delta stuff is sold in two parts: rough-in valve, then a second package with the trim and cartridge. Big box stores may package those two together, but they give you fewer options.
The Delta R10000 rough-in valve would allow you to choose between three different types of valve functions depending on which trim/cartridge you choose (and, you could change your mind 10-years down the road without changing the rough-in, only the trim and cartridge!). Those three choices are: single handle pressure balance; two controls - one for volume, second for temperature; two controls - one for volume, second for thermostatic control. The price goes up as you go in the string I mentioned, partly because of the additional moving parts and partly because of the more complex trim. Then, depending on the finish on the trim and the style, those throw in different price points.
In reality, though, if you stick with one of the major brands, most work pretty well. All code compliant ones will have anti-scald technology that your old valve probably does not have, and may also be available in no-lead versions (which is becoming a requirement).