Hello & welcome,
What your are calling as a subcutaneous cyst
which is present on lower lip, i think, is a mucocele
& lower lip is the most obvious site for mucocele to occur.
It occurs due to injury
to salivary glands or their ducts thus damaging or destructing them to the extent that ducts are not able to carry any formed saliva from gland to oral cavity for which they are meant to be.
But, production of the saliva continues as usual and so it starts accumulating under the lip mucosa thus giving it a swelled appearance.
It is harmless unless infected secondarily. It tends to come & resolve after some time . This continues intermittently.
Mucocele , present in any location on lower lips should be removed surgically with thorough enucleation or removal of associated salivary glands.
Most prominent feature that mucocele presents with is recurrence even after removal within a few months or a year. Recurrence rate among the mucocele is very high.
So, special care has to be taken while doing surgical removal of mucocele & that is no remnants of the associated salivary glands should be left behind so as to avoid its recurrence.
Antibiotics might be necessary as if mucocele is infected secondarily & should be associated with painkillers & anti inflammatory drugs to reduce pain & discomfort related to it.
Avoid biting or giving pressure on it with tongue as this may spell the con-taints out leaving it more prone to recurrence. Visit your dentist
& get it removed as soon as possible.