Most throw rugs and runners get bought by looking at the room. Dad doesn't walk the room. He walks a line: bed to bathroom, chair to kitchen, back door to the hall. The rug that matters is the one that sits on that line, so the piece has to be sized to the line.
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Walk the route once, then measure it
Pick the two or three routes he uses every day, especially at night and first thing in the morning. Stand at the start point and walk each one at his usual pace, the way he actually moves, with or without a walker, and with or without a bag in his hand. Notice where he reaches out to steady himself. Those spots need the most even floor.
Measure the clear width of the path at its narrowest point, wall to wall or wall to furniture. Then measure the length the runner has to cover, from the start of the route to the end, plus enough room for a foot to land cleanly at each end. Don't count space in the path of a door swing as usable floor. If a door opens over the rug, the door will catch the edge every time it moves.
Check the edges and the floor under them
A runner is only as safe as its edges. Look at the floor on both sides. A raised threshold, a change in flooring, or a cord running under the edge all create a lip that a walker tip or a slipper can catch on. Where the runner meets a doorway, the edge should lie flat against the threshold. If it can't lie flat, leave that doorway without a runner.
Ask whether the backing is meant for your floor. A backing that grips wood may do little on tile, and a backing made for carpet may slide on hard floors. If you're keeping a rug that slips, put a non-slip pad under it sized about an inch smaller than the rug on every side, so the pad never shows at the edge and never sticks out past it.
Decide what to remove, not only what to add
Often the right answer is fewer rugs. Walk the house and ask of each one whether it sits on a route he uses. A decorative piece he crosses only to sit down is the first candidate to go. Taking up a rug is a floor decision, and you can present it to Dad as one: you're clearing the walkway, not judging the living room.
What to compare before you buy
Compare on three things, and let them decide before the color or the sale tag does: the width that matches the path you measured, the backing that matches the floor you have, and whether the edge can lie flat at every doorway on the route. Ask about the return policy before you buy, because the first size is often off by an inch once the route is measured for real.
Next move
Tonight, walk the route with a tape measure. Write down the narrowest width and the total length, plus the door swings that cross the path. Then look at the rug already on that line, check its backing and its edges, and take those numbers with you to the store. If you're starting from zero on the rest of the house, the Start Here page sets the order.