Just a few idea; Weather stripping, Hot Water Heater (HWH) Blanket, check the temp on the HWH, See if there is energy saving modes on the washer and dryer and show her how to use them, programable thermostat, and maybe the turbines or vents (don't know what there called) in the roof to let the heat out in the summer if they don't have um.
Touch the top surfaces of the tank (not near the piping or relief valve) or shoot them with an IR gun, then compare to other surfaces around you. If it's a few degrees warmer and it's an electric tank, a blanket or a new heater would help although I'm not sure how much. The temp differential of the tank is easier to assess in the winter time when the room is cooler.
Most newer tanks are very good designs.
A simpler response is just to feel the warmth of the tank. If it feels very warm, then it is leaking BTUs.
For NG tanks it's a bit more difficult since they have that big heat wick in the center and the blanket should not get to close to the flue.
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Not knowing your climate and thinking like a Canuck...
If it's going to be winter when you get back, with a $100 budget I'd do the following (obtain an IR temp gun - maybe $50 or borrowable) and scan the house from the *outside* on the first really cold night. Consider expanding foam where you find big heat leaks or use other forms of insulation depending on what you are dealing with. Use this gun to help with determining where to put weatherstripping as well. It's all about slowing the heatloss from the house.
A programmable t-stat can often be had cheap (mine was $30 plus I got $15 rebate from the gas company). A few caveats. The savings will be when it flattens out to the lower temp, so don't go too wild or your just putting too much heat up the flue (if the heating system burns oil or gas) on recovery. If you have water heated rads and piping that is exposed, be careful that it doesn't have time to freeze when the temps get set back.
If there isn't a "solar powered clothes dryer" installed, set up a clothesline outside. ;-)
Weather stripping and window film to block out some heat radiation so you can keep your curtain/blinds (whatever) open and not use those CFL's Depending on how many windows you have (and their direction) that will probably put you around $50-$75. Spend the rest on a programmable T-stat (I bought mine for $20 used off eBay, works flawlessly).
Time is the best teacher. Unfortunately it kills all its students.
put a timer on the hot water heater, the water will stay hot for 2+ hours, and warm for 4+ hours, so if your mom is not the kind of person to take long showers at mid night, and she has a day job have the water heater come on about an hour befor she normaly wakes, and turn off about the time she goes to work, then turn on again shortly befor she gets off from work, and turn off a few hours befor bed time, this can cut the energy used by the water heater in half.
with the rest of the money get pipe insulation for the hot water pipes, and a bag of spray can foam to fill all the gaps around the sill plate of the house, around doors, windows, and pipes, if you notice a window is drafty pop off the trim with a flat pry bar and fill the gaps with "door and window" expanding foam, as it is softer and will not bow doors and windows, but seals them nicely.