- The Master to the Doctor (The Sound of Drums)
First let us look at the events that follow The End of Time with respect to the Master. We know from The Sound of Drums that he ran away in the initial days of the Time War and thus was not there to witness the Doctor burn both Time-lords and Daleks with the Moment. The conclusion of The End of Time leaves him stranded and back on Gallifrey. I believe he regenerates into Missy on Gallifrey by either his own choice (maybe he steals regeneration energy from a Time-lady) or as a punishment from the Time-lords. Whatever may be the case, what we can definitely say is that he is present on Gallifrey when the events of The Day of the Doctor take place.
- Missy to the Doctor (Death in Heaven)
Once out of the Time-lock she decides to give the Doctor a present on his Birthday. She goes through his timeline to study him and the people surrounding him, so that she can get him the ultimate present. By this time I think the initial phases of her plan have already begun. Remember the Half-Faced Man from Deep Breath and Gretchen from Into the Dalek, the version of Nethersphere we see then (maybe a prototype) is very different from the version we see in The Caretaker. The former had a more outdoorsy feel to it. I think that the Cybermen hadn't really come into play by that time and Missy was only using the Matrix to capture the recently deceased and have a rendezvous with them. After which I think she discovers that she could use this afterlife business to make him an army of Cybermen. Gift decided. But birthdays can be very tricky, a lot of effort is required to create that shocking sense of surprise. More precisely speaking, Missy now needs the Doctor to be at the right place at the right time. This is where Clara comes into play.
- The Doctor to the Master (The End of Time)
We also know that the Doctor chose not to kill the Master in The End of Time. I think this simple act of kindness causes the rekindling of the fire of friendship in the hearts of Missy. When she comes to know of the War Doctor's struggles, of how the Doctor had to abandon his very own ideals due to the Time War, she empathizes with her old friend and decides to re-establish their friendship but in her own psychotic way. She decides to show the Doctor that they aren't different given that they have the power to conquer universes. She then escapes from Gallifrey by stealing a TARDIS. How she escaped from the Time-lock is still a big question. My theory is that she uses the same passage the Doctors use to enter the Time-lock, just for the opposite purpose.
- Missy to the Doctor (Death in Heaven)
Missy needs to control the future events in such a precise way that he ends up on Earth on his birthday and in dire need of an army that could help him stop the bad guys. She decides to be the bad guy, so that she can make him understand what he really wants. She also needs someone who would make the Doctor search for heaven/hell/afterlife, someone with so much influence on his life that he would go the distance. We know that time can be changed with precision but there are certain events that never change. I think Clara was always destined to meet Danny and in every possible future Danny was destined to die. And I think this is why Missy chose Clara. With Clara she has the liberty to manipulate past events and still end up with Danny dying. Of course then she could use her control freakiness to manipulate the Doctor into the looking up 3W, that too on his birthday.
- The Doctor to Clara (Dark Water)
So Clara it is, but why would the Doctor even care for her? Why would he decide to do something for her that even the TARDIS will oppose (he switches off the TARDIS safeguards before looking for Danny)? The answer is of course that he won't, not unless Clara becomes someone very important in his life. What happens next is a bit complicated so just bear with me. There are two timelines when it comes to Clara. The timeline where Missy gives her the Doctor's number is the original timeline. Clara then calls the Doctor and the events of Series 7B take place. In The Name of the Doctor when Clara jumps into the Doctor's time stream an alternative timeline is established, the one with Clara's echoes which in turn cause the events of Asylum of the Daleks and The Snowmen. Her sacrifice clearly strengthens their bond and makes Clara an important person in the Doctor's life. Clara helping the Doctor realize that Gallifrey can still be saved in The Day of the Doctor only adds more strength to their bond.
I believe that Missy knew that the events of The Name of the Doctor will course-correct in The Time of the Doctor otherwise she might have confronted the Doctor earlier. Also after the events of The Time of the Doctor the alternate timeline reverts back to the original timeline, the one where the Clara-echoes don't exist. To all those who are outside the influence of the alternate timeline, it appears as though the events of The Name of the Doctor never happened, but The Doctor, Clara, Paternoster Gang and I believe even Missy(owing to the fact that she is a Time-lady) remember about them. Thus it is important to remember that the events of the original and alternative timeline form a loop which still leads to Danny dying in Dark Water. After which the events of Dark Water and Death in Heaven follow.
- Missy (Flatline)
So this is why I think Missy chose Clara. In conclusion I think Clara as a companion was her invention from the start and everything that has happened to the Doctor and Clara since they met in The Bells of Saint John has been an indirect result of Missy's involvement. It's obvious that she miscalculated the Doctor completely because we now know that the Doctor is not an officer for an army, in fact he's just a wonderful idiot with a blue box and screwdriver.
What sort of a person goes through so much trouble just to give a present? you might ask. I will answer this by using the exact words that the Doctor used to describe the person who put the advert in the paper in Deep Breath a.k.a. Missy. An egomaniac, needy, game-player sort of person.
Of course this theory makes you to wonder if this is how Moffat planned it all along. If that is indeed the case then I think we are in for even more bizarreness in the future.