Jon Snow would somehow have had to brush up on ancient and mystical
knowledge on the Children of the Forest's symbology in order to
make these seemingly accurate drawings.
These symbols you see bear significance in the culture of the
Children of the Forest, which in turn is reflected in the White
Walkers (which are created by the Children of the Forest)
The spiral cave painting
here on the left coincides with the same geometric pattern that the
White Walkers put together using corpses in Season 3.
pattern, here seen in a top-down view of one of an elaborate monument
made by the Children of the Forest, where we have Stonehenge-style
standing stones in a spiral pattern, seen in S06E05. The creators of the show talk about this pattern in this scene on 'Game of Thrones Season 6: Inside the Episode #5'
Here we have what looks
like the Greek symbol, phi ('Φ'), on the left coinciding
with the same geometric pattern of corpses that the three rangers in
S01E01 run into, also put together by White Walkers.
The only Westerosi people who might know this much about the White Walkers, Children of the Forest and their symbology are Sam Tarly, Bran Stark (Three-eyed Raven) and perhaps some members of the Reed family. Neither Jon Snow nor Davos would know these things. While the Night's Watch and the Wildlings are aware of these symbols made by the White Walkers, they wouldn't make the connection with the Children of the Forest, so for Jon Snow to paint these symbols and attribute them so confidently to the Children, while being very accurate, seems extremely far-fetched.
This whole idea of Jon Snow faking the cave paintings is essentially debunked, implicitly, by the creators of the show in 'Game of Thrones: Season 7 Episode 4: Inside the Episode (HBO)', where D&D discuss the importance of the symbols in the cave and how it relates to the Children of the Forest and the White Walkers.
Jon Snow would have to have had blue paint on-hand
Close-up of the cave
painting of the Night King and his White Walker lackeys. The blue in
their eyes stands out as quite intense and vivid.
Blue is historically the most difficult colour to make pigments
and dyes and paints of, in our world.
If we assume the fan-theory to be true, then we also have to assume Jon
Snow brought blue paint with him to Dragonstone, just in case he
needed to draw some White Walkers in a cave, just in case Dany didn't
believe him. It's not impossible, but not very believable.
Incidentally, on the topic of blue, I found this quote on
Reds, blacks, browns, and ochres are found in cave paintings from the Upper Paleolithic period, but not blue.
—Blue (In the
ancient world), Wikipedia
It's an interesting, funny fact that would coincidentally lend
credence to the fan-theory, but one shouldn't compare non-fictional,
early Homo Sapiens from the Upper Paleolithic era with fantastical,
magical, spell-casting Children of the Forest in a fictional setting,
in order to arrive at conclusions about the availability of pigments
for cave paintings.
Jon Snow would have to be a skilled liar
Jon is honourable and true (not a trickster or liar), much like his
surrogate father, Ned Stark.
The last time Jon specifically lied was when he had to deceive the
wildlings into believing he was a turncoat crow - he even killed
Qhorin Halfhand to sell this deception.
The thing is, he was already sexually attracted to Ygritte and was
starting to have conflicting feelings about the wildlings, and the
pointless violence between them and the Night's Watch, and he was
starting to see that what he was told about the wildlings wasn't
true, after spending some days with Ygritte. Had he not had these
conflicting feelings, he would have executed her when he had the
chance. All this to say: he didn't really lie, so much as he was
following Qhorin's orders, and leaning into his true, new, growing
feelings about the wildlings.
It would be very unlike Jon Snow to lie to Daenerys about this, not to mention the risk of getting caught lying, whether because he
was spotted by guards carrying paint around, or spotted in the middle
of painting, etc. would jeopardize the entire mission (the most important mission in the world) of getting help for the War for the Dawn - a very high risk.
This is one of those fan-theories where it sounds funny/entertaining at first glance, but the more you think about it, the more unlikely it sounds.