It’s very simple to re-sod a brown grass patch. The real keys lies in preparing the soil before you lay the new sod.
Step 1 - Removing Old Sod
Dig around the perimeter of the brown grass patch, down to a depth of about 3 inches. Remove the old sod and lift it out, being careful to get the roots, too. In most instances you can compost this. The only exception would be if it had been damaged by insects.
Step 2 - Preparing the Soil
Use a pitchfork to loosen the soil in the brown grass patch. Dig it deep and turn it over. Fill with new soil (you can also use compost) so until it’s only about 1 inch to 2 inches below the level of the surrounding grass. Rake it until it’s flat and then water heavily. Give the water time to be absorbed by the soil.
Step 3 - New Sod
Make sure you new sod matches the type of grass you already have in the area. This will give an even finish to the lawn. Trim the new sod and lay in the hole you have. Where it doesn’t fit exactly, fill in with soil. You’ll initially need to water very regularly, probably every day in summer, until the new sod has become established in the old brown grass patch.