Your deductible is the piece of a claim you cover before the insurance company covers anything. Raise it and the premium comes down, because you've taken back some of the risk.
On auto, moving from a $500 deductible to $1,000 commonly trims comprehensive and collision premium by a meaningful percentage. On homeowners the effect is usually bigger, because the deductible jumps are bigger. Your agent can quote it both ways in about two minutes, and you should make them, since the savings varies a lot by carrier and by where you live.
Then do the division. If raising the deductible by $500 saves you $120 a year, you're ahead after four claim-free years and behind if you file in year one. Most households go many years between claims, which is why this usually pays. Most is not all.
The condition is cash. A higher deductible only works if you can write that check on a bad Tuesday without touching a credit card. If your emergency fund is thin, the lower deductible is the better buy even though it costs more on paper, because you're paying the insurer to hold cash you don't have yet. Build the fund first, then raise the deductible, in that order.
Two things to check before you change anything. Homeowners policies in some areas carry a separate percentage deductible for wind or earthquake, which is a different and much larger number than your standard one. And a low deductible does nothing for the claim you actually fear, since the expensive losses blow past any deductible you'd pick.
One more move in the same family. Raising your deductible lowers the premium on the small claims you can absorb, and the money you save is often better spent buying more liability coverage, which is the part that protects you from a claim you can't absorb. Cheap to add, and almost nobody looks at that number.
How much cash makes the higher deductible safe is the same question as how big your emergency fund should be.