The sound comes first
The first sign is usually a sound. You turn the key and the engine does not turn over the way it used to. It labors, slow and tired, or it makes a quick clicking, like a pen tapping paper, and nothing else happens. Some mornings it starts fine, and that is what makes it easy to explain away. Cold weather is when a weak battery shows itself, because cold makes the battery work harder for the same start.
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Other early clues: the dashboard lights look dim with the engine off, the headlights are weaker than they were, the power windows move slowly, or the radio resets when he starts the car. Any one of these alone may mean little. Two or more together, on cold days, deserve a check.
What to check at the car
Look at the date sticker on the battery, which is usually on the top or side of the case. Its code tells you roughly when it was made. An old battery is not automatically a dead one, but it is one to watch closely. Check that the terminals are clean and tight, with no crust of white or green powder and no clamp that moves when you pull it. Corrosion alone can cause a no-start.
Ask how long the car sits between drives. A car that goes days or weeks without running slowly drains its battery, even with nothing left on. If it sits a lot, plan a regular drive, or ask a shop about a charger that keeps the battery topped up while the car is parked. Be careful about what gets switched on without the engine running, since headlights and the radio pull from the battery directly.
What to ask the shop
Ask the shop to test the battery and the charging system, not just to replace the battery. A failing alternator can drain even a new battery, and you would end up paying twice. Ask them to show you the test result and explain what the reading means for the coming winter. Ask whether they can write the result down so you can compare it the next time you visit.
What to do tonight
Before the next cold morning, sit in the car with the engine off, turn on the headlights, and watch them. Then try the key once. If the lights dim badly or the starter only clicks, stop. Repeated attempts drain the battery further. Call for a tow or a jump start, and plan the shop visit for the daytime.
This is the kind of job that is easiest to catch while it is small. Your father will probably say the car has always started fine, and he may be right right up until the morning it does not. Say so gently, and offer to be there for the test.