How can I repair the current material (stone) under this pergola?

Choose some earthy colors and paint the stone. For ideas go somewhere that sells stone as see what colors they are naturally come in. Probably a moss green, faded blue, perhaps a light or very faded pink, etc. Just leave the grout lines unpainted for a natural look.

Have you tried pressure washing just to brighten it up and then you could make repairs where needed?

I would power wash, paint and then seal it with an outdoor poly. It's in good shape.

First power wash then i would get some pebble stone and put in between openings then apply sealant

power wash and if it reveals dirt between the stones, add moss.

if you don't want plants in the stones, go with boiling water and vinegar to kill the weeds off. Otherwise, maybe some big ceramic pots with bushy plants/trees to add shade?

Pressure washing is a great first step.If you want to get a really clean look, dig out a 1/4 to half inch of whatever is between the stones and add pure white limestone paver base on top of the material in between. And moss is a great idea. In succeeding seasons if the paver base gets dirty-looking, moss will grow in the paver base. We got our moss without invitation, it just grew there, very pretty.

Powerwash and then put colored sand between the stones.

Sounds like everyone wants you to powerwash and I am thinking - budget! So don't rent a powerwasher and use you hose. If you hold it at an angle and get close you should be able to do it ok. But then it will just go back to looking the same after a while, right? So you have to either keep washing it or spend the money for a sealer.

PLease let us know what you do!

you can get a blue stone cement colorant at lowes, it costs about 8$ and a little goes a long way. just spray it in a garden sprayer, and then spray some wi9th a little water to get natural stone variations, then seal it with a quality concrete sealer. it will look like natural blue stone when you are finished. good luck.