So you've decided that bike commuting is for you. Great! It doesn't have to work every day, but for the days it does, you are going to be maximizing that time spent pedaling.
How you value your time changes depending on what your values are. It depends on how much money you make and your time constraints. Let's say you value your time at $25 per hour. A bike commute takes 25 minutes and a car commute takes 12 minutes each way. Ignoring all the countless cost/benefit trade-offs, that's $10.83 worth of productive time you are spending in the saddle. So, let's try to recoup that $10.83 through the commute itself.
One biggest thing you can do is to cancel your gym membership and just ride your bike for exercise. This is huge if you mainly use the gym for cardio. For a $20 gym membership, that comes out to ($240 per year / 52 weeks per year/ 5 work days per week) $0.92 per commute day. Another thing you can do to add value is to listen to educational podcasts or audiobooks. These could be to prepare you for a test, help you achieve a goal, or just keep you better informed. You could also listen to these in the car, but when biking you get more time to do so. This adds up to (13 more mins per commute * 52 weeks per year * 5 work days per week) an extra 56 hours of education per year, the equivalent of a semester-long 3 credit university lecture. If we value this time as such, it would be worth about $850 based on 2018 average US tuition rates, or $3.27 per commute day.
Biking really isn't conducive to eating out for lunch as your range is fairly limited for the lunch hour. Bringing and preparing a large lunch can be a big time commitment but is doable. Some people use batch cooking where they cook all of their meals on Sunday then freeze them. Rather than bring and make a large lunch, I would rather wake up 15 minutes earlier to make and eat a large hot breakfast. Breakfast foods (eggs, milk, coffee, bacon) are usually the most affordable items you can buy at the grocery store. After eating this large breakfast, not only can you reduce the cost of lunches by just eating small dry snacks to finish your work day, you gain 30 to 45 minutes of productivity by taking a shorter lunch. This method nets 30 additional minutes of productivit, and saves on food costs. This is possible to do with a car as well, but removes the temptation to eat out, so you will have more successful repetitions of this lunch method per year. So, assuming if you drove, you would eat out once per week, and if you bike, you would eat out .25 times per week (about once a month), this comes out to ((30 minutes * $25 per hour) plus ($10 lunch minus $3 breakfast) )*(52 weeks / (52 weeks *.25)) $2.92 per commute day.
Car costs can get a little subjective, so to keep things simple here, lets just use the IRS mileage rate to account for the gas and wear and tear saved by not driving. This commute is going to be 6.25 miles so at current IRS rates, this gets us to (6.25 2 .572) $7.15 per commute day.
so, the savings of commuting by bike this distance comes out to (7.15+2.92+3.27+0.92) = $14.26 per commute. With the cost of additional commute time being $10.83 per commute, this leaves us with an additional $3.83 in our pocket per commute day. Though your circumstances may be different than described above, it goes to show that bike commuting within 6.25 miles can be an effective way to cut your costs, even if you own a vehicle.