Cut off leafy tops and peel tough outer layers from lemongrass. Trim off and discard any discolored or dry parts of root ends. Cut trimmed, woody sections into 2-inch lengths; crush lightly with a mallet. Add lemongrass and ginger to honey. Place over medium-high heat in a 3- to 4-quart pot; stir occasionally until honey simmers, about 5 minutes. Simmer 10 more minutes; remove from heat and let stand until cooled slightly. Pick out and discard lemongrass; cool completely. Pour into clean decorative jars, cover and store at room temperature up to 1 month.