How do I easily and inexpensively fix a crack developing in the stone front of my house?
Your picture IS worth a thousand words! Is the stone a fascia or veneer in front of the bearing wall? Or is the stone the actual foundation for the structure? That's a window sill in the photo? Lastly, how old is the stonework? If it's just a façade of stone you may be able to reason the crack developed from settling. But it would be wise to monitor it. But to repair the gap start by removing loose pieces by hand (specifically that one in the middle). Then check each stone is still fastened and secured in place. Do this by grasping and attempting to move each one (again by hand). The second stone from the top left might come loose. If any are loose get a batch of mortar ready. With a cold chisel Gently chip away the old mortar from the joints on either edge of the crack. The idea is to develop a space along the crack line for fresh mortar. It will be difficult, but try not to loosen any stone. Brush or vacuum  out any dust. You should now have a wider gap running the length of the old crack! With a bucket of water and a sponge moisten the surrounding area. Mix a batch of mortar with enough water so that it will remain in place when troweled into the gap. The ideal tool for this type of work is a 'joint striking tool' and a pointed trowel. Work your way up from the bottom pressing mortar in the gap with the pointed trowel until it is full. Every 6 inches or so switch to the joint striking tool (which is concave in profile) and press the still moistened mortar level with the surrounding original mortar. For best results let the mortar cure out of direct sun light so it doesn't dry to fast. After it has set up for a couple of hours keep it moist for the same reason. The slower it cures the stronger it will be. Mortar straight from the bag and mixed with enough water will usually appear much lighter when cured. Some what silvery to grayish in tone. The mortar in your photo has been wiped with a sponge. Probably to clean any splotches from the stone and/ or for appearances. You can do the same after it sets up in an hour or so.