In fact, the native speakers (5 in total) have no real idea either and were only able to make guesses.
1. A couple of versions based on some f.acc.sg. form of an adjective/participle:
- "the fallen one" (but why exactly "під-"?..);
- "the light(-feathered) one" (pretty far-stretched).
2. "The gun" was also actually mentioned, but it has many problems (most importantly, being completely unattested somehow, which is highly suspicious from the typological perspective).
3. The version which seems most convincing to me so far is it's a dialectal form of "under (one's) arm" (= "під пахву", lit. "under the armpit"). It was, in fact, proposed by a Russian scholar, who has pointed at the existence of the West Russian dialectal word подпали "armpits"; phonetically, semantically and morphologically it's flawless, the only problem is it isn't directly attested (but "an armpit" is much, much more likely to have an unattested dialectal form than "a firearm").