Does it make sense to talk about origin of self? In dependent origination at what stage self comes into picture ? Does self originate ?
In dependent origination, the stage self comes into picture is at 'attachment' and is full mature at 'birth'. As a modern Indian, you (the questioner) should know the word 'jati' ('birth') does not mean 'physical birth from the mother' but means 'self or social identity'.


  Jāti (in Devanagari: जाति, Bengali: জাতি, Telugu:జాతి, Kannada:ಜಾತಿ,
  Malayalam: ജാതി, Tamil:ஜாதி, literally "birth") is a group of clans,
  tribes, communities and sub-communities, and religions in India. Each
  jāti typically has an association with a traditional job function or
  tribe. Religious beliefs (e.g. Sri Vaishnavism or Veera Shaivism) or
  linguistic groupings may define some jatis. A person's surname
  typically reflects a community (jati) association: thus Gandhi =
  perfume seller, Dhobi = washerman, Srivastava = military scribe, Ror =
  Warriors etc. Wikipedia


The self certainly originates, as described in many suttas below: 


  And what is clinging? These four are clingings: sensuality clinging,
  view clinging, precept &amp; practice clinging, and doctrine of self
  clinging. This is called clinging.
  
  SN 12.2
  
  
  
  The craving that makes for further becoming — accompanied by passion &amp;
  delight, relishing now here &amp; now there — i.e., craving for sensual
  pleasure, craving for becoming, craving for non-becoming: This, friend
  Visakha, is the origination of self-identification described by the
  Blessed One.
  
  MN 44
  
  
  
  There is the case where an uninstructed, run-of-the-mill person — who
  has no regard for noble ones, is not well-versed or disciplined in
  their Dhamma; who has no regard for men of integrity, is not
  well-versed or disciplined in their Dhamma — assumes form to be the
  self. That assumption is a fabrication. Now what is the cause, what is
  the origination, what is the birth, what is the coming-into-existence
  of that fabrication? To an uninstructed, run-of-the-mill person,
  touched by that which is felt born of contact with ignorance, craving
  arises. That fabrication [of self] is born of that. And that fabrication is
  inconstant, fabricated, dependently co-arisen. That craving... That
  feeling... That contact... That ignorance is inconstant, fabricated,
  dependently co-arisen.
  
  SN 22.81
  
  
  
  This, monks, is the path of practice leading to self-identification.
  One assumes about the eye that 'This is me, this is my self, this is
  what I am.' One assumes about forms... One assumes about consciousness
  at the eye... One assumes about contact at the eye... One assumes
  about feeling... One assumes about craving that 'This is me, this is
  my self, this is what I am.'
  
  "One assumes about the ear...
  
  "One assumes about the nose...
  
  "One assumes about the tongue...
  
  "One assumes about the body...
  
  "One assumes about the intellect that 'This is me, this is my self,
  this is what I am.' One assumes about ideas... One assumes about
  consciousness at the intellect... One assumes about contact at the
  intellect... One assumes about feeling... One assumes about craving
  that 'This is me, this is my self, this is what I am.'
  
  MN 148
  
  
  
  Who, O Lord, has a sense-impression?"
  
  "The question is not correct," said the Exalted One.
  
  "I do not say that 'he has a sense-impression.' Had I said so, then
  the question 'Who has a sense-impression?' would be appropriate. But
  since I did not speak thus, the correct way to ask the question will
  be 'What is the condition of sense-impression?' And to that the
  correct reply is: 'The sixfold sense-base is a condition of
  sense-impression, and sense-impression is the condition of feeling.'"
  
  "Who, O Lord, feels?"
  
  "The question is not correct," said the Exalted One. "I do not say
  that 'he feels.' Had I said so, then the question 'Who feels?' would
  be appropriate. But since I did not speak thus, the correct way to ask
  the question will be 'What is the condition of feeling?' And to that
  the correct reply is: 'sense-impression is the condition of feeling;
  and feeling is the condition of craving.'"
  
  "Who, O Lord, craves?"
  
  "The question is not correct," said the Exalted One. "I do not say
  that 'he craves.' Had I said so, then the question 'Who craves?' would
  be appropriate. But since I did not speak thus, the correct way to ask
  the question will be 'What is the condition of craving?' And to that
  the correct reply is: 'Feeling is the condition of craving, and
  craving is the condition of clinging.'"
  
  "Who, O Lord, clings?"
  
  "The question is not correct," said the Exalted One, "I do not say
  that 'he clings.' Had I said so, then the question 'Who clings?' would
  be appropriate. But since I did not speak thus, the correct way to ask
  the question will be 'What is the condition of clinging?' And to that
  the correct reply is: 'Craving is the condition of clinging; and
  clinging is the condition of the process of becoming.' Such is the
  origin of this entire mass of suffering.
  
  SN 12.2