Slice cucumber in half lengthwise, then scoop out the seeds. Slice lengthwise in half again, then slice thinly with a mandolin or knife to make small quarter slivers.
Mix cucumber slices with a teaspoon of salt. Place cucumbers in a small strainer over a bowl. Let sit for 30 minutes.
Squeeze as much excess water as you can out of the cucumbers, so your sauce isn't too runny.
In a bowl, add sour cream, cucumber slices, dill, garlic, lemon juice, and salt, and mix. Let sit in refrigerator for at least a few hours for the flavors to blend properly, then taste and add a pinch of salt if necessary.
Homemade tzatziki sauce will keep in the fridge for up to 5 days.