You want a method that doesn’t involve clocks, because belief in clocks is belief in the authority that says what a “good” clock is, and what it measures. You are trying to get your normal person to “believe because it’s true”, not “believe because clever people tell you to”.
Your only method is therefore to find an annual phenomenon and tell your friend to count the days between one appearance of it and the next. If after four repetitions he has counted 1461 days, you will have proved your point.
The Stonehenge method involves measuring the direction of sunrise at the solstice - for instance, at midwinter, “When does the Sun rise furthest south?”. You can do this by standing reasonably far from a trilithon and as the sun rises, moving rapidly left or right until the rising sun is exactly framed by the stones. Then push a stick into the ground and come back tomorrow and do it again.
If the solstice occurs near the time of sunrise then one day will clearly be “the winner” in all this. If the solstice occurs half way between two sunrises then they will be “joint winners”. Either way, you will have an approximate time of the solstice and repeating the experiment over a few years will give you a good estimate of the time, in days, between one solstice and the next.
The meridiana method was used from the 16th to the 18th century. Bore a hole high up in the south wall of a church and dig a trench in the floor going due north from that point. Measure the position of the patch of light at noon each day. You get noon from the fact that the trench goes due north; you get the horizontality of the trench from filling it with water during construction; you get a precise north by careful observation on the day of a solstice.
This method is so sensitive that it ends up being useful in detecting movements of the building itself in the first few hundred years of its existence - which is why on one occasion the Baths of Diocletian were used, as being 1500 years old and not likely to move much.
With this equipment it was possible to measure the length of the year; the inclination of the ecliptic; and many other astronomical data. J.L. Heilbron’s The Sun in the Church is the absolutely definitive scholarly reference on all this.
The amateur method is to recognise that it doesn’t have to be solstices: any repeatable time of year can be used. In March and September the position of sunrise moves about 30 minutes of arc per day - that is, the whole width of the Sun each day. That is easy to observe. It moves quite fast even in October. So:
- Tomorrow morning, at dawn, place yourself so that the rising Sun is just obscured by something - a tree-trunk or a tall building. “Just obscured” because that is the easiest phenomenon to observe. Drop a splash of yellow paint on the ground at this spot.
- Do the same for the next four days. You will see that the splashes are pretty far apart.
- Next year, do the same but with green paint. You will see that each green splash is near the corresponding yellow splash from 365 days ago but not actually on it. It will indeed fall short by about a quarter of a day - which will demonstrate “365¼” to your friend.
- If your friend is obstinate and you are patient, you can do this three more times, with red blue and white paint, until you see that the white splashes do almost exactly cover up the yellow splashed from 1461 days before.