Cut the ginger in half, set one half aside. Peel the remaining half, and either slice into large medallions (if you want to fish the pieces out), or mince.
In a large saucepan off the heat, toss the rhubarb, ginger, sugar, and lemon juice. Let sit about 10 minutes, or until it has released juices.
Stirring occasionally, bring to a boil over medium-high heat. Reduce heat to lower it to a simmer and cook to let it partially break down, about 5 to 7 minutes. Stir occasionally. Remove from heat when done.
Slice remaining ginger into quarter or half inch chunks, then squeeze in a citrus juicer to release liquid. You should end up with about 1 tbs. Discard solids, then stir juice into the mixture.
Let cool completely, then enjoy with a spoon, on top of ice cream or yogurt, or baked into crumbles.