Cut off lemon ends. Cut lemons lengthwise into quarters, then cut crosswise into thin slices, removing and reserving seeds. Place lemon seeds in small bowl. Place lemon slices in medium bowl. Add 1/2 cup cold water to bowl with seeds and 3 cups cold water to bowl with lemon slices. Cover bowls with plastic wrap and let stand 24 hours at room temperature.
Transfer lemon slices with their soaking water to heavy large saucepan. Strain water from bowl with seeds into same saucepan. Wrap seeds in cheesecloth; tie with string and add to saucepan. Bring to simmer over medium-high heat. Cover partially and adjust heat so mixture barely simmers. Cook 45 minutes, stirring occasionally.
Puree sliced ginger with remaining 1/4 cup water in processor, stopping occasionally to scrape down sides of bowl. Strain mixture through sieve, pressing down on solids with spoon. Reserve 1/4 cup ginger juice.
Remove cheesecloth bag from saucepan and squeeze it between spoons so liquid drains back into pan. Add 4 cups sugar to lemon mixture and stir until dissolved. Add reserved 1/4 cup ginger juice. Simmer mixture uncovered until it reaches gelling stage, about 1 hour. (To test for doneness, remove pan from heat. Fill chilled spoon with preserves, then slowly pour preserves back into pan; last 2 drops should merge and sheet off spoon. One tablespoon of preserves spooned onto chilled plate and frozen 2 minutes should wrinkle when gently pushed with fingertip.)
Rinse clean jars, lids and screw bands in hot water. Spoon preserves into hot jar to 1/4 inch from top. Immediately wipe rim, using towel dipped into hot water. Place lid on jar, seal tightly with screw band. Repeat with remaining preserves and jars. Arrange jars on rack set into large pot. Cover with boiling water by at least 1 inch. Cover pot and boil 15 minutes.*
Remove jars from water bath. Cool to room temperature. Press center of each lid. If lid stays down, jar is sealed. Store in cool dry place up to 1 year. Refrigerate after opening. (If lid pops up, store preserves in refrigerator.)
* If preserves have not been processed in water bath as described above, cover and refrigerate.
Makes about 3-1/2 cups.
Recipe from Bon Appetit, October, 1991.