Why does President Obama use "ISIL" to refer to the Islamic State?
It's not because he's secretly a Muslim or ISIS sympathizer. First, why would he advertise or telegraph it with such a clue? (See: "Secret") Second, Muslims in the MidEast call the Islamic State "Daesh", which is the group's Arabic acronym and a pun - it sounds similar to an Arabic word for a racist who forces his view onto other people.
Yeah, the Arabs are calling the Islamic State "Troll Nation."
Finally, ISIS refers to themselves simply as "The Islamic State", or at most ISIS, so the theory that the President is signaling with ISIL just doesn't make sense.
So, why ISIL and not ISIS like everyone else?
We need to review the history a bit to discover the reason. When ISIS was just another Syrian rebel group fighting for scraps of territory, they called themselves ISIL, "The Islamic State in the Levant". The Levant is an old name for the region surrounding east of Israel, and the name was aspirational in nature - it's what they wanted to become. After they conquered large parts of both Iraq and Syria, they updated their name to ISIS, "The Islamic State of Iraq and Syria."
This actually matters a great deal. The President holds as a great foreign policy achievement ending the war in Iraq. Ending the Iraq War was a cornerstone of his 2008 candidacy, and a capstone of his 2012 re-election campaign. So referring to the Islamic State of Iraq and Syria would be an acknowledgement that his foreign policy failed in exactly the way his critics predicated - creating a power vacuum filled by radicals and Iran.
A large portion of the President's recent foreign policy and rhetoric has been built around denying, ignoring, or avoiding facts and realities the present challenges too or contradict his messaging. It it the President's denial of the fact that a massive amount of Iraq has fallen to ISIS that leads him to avoid using the word ISIS. (And Daesh, as it's essentially the same thing in Arabic)
This evasiveness has reached absurdest levels recently, but was a feature of this administration from the beginning. It was the source of the attempt to rebrand terrorist attacks as "man-caused disasters" and calling the Fort Hood shooting an act of workplace violence. Even now, the President and the administration refuse to call terror attacks by Muslims Islamic terrorism, as that would be an admission that his broad-ranging first-term outreach utterly failed. This administration has, from the beginning, attempted to eliminate hard truths that they don't want to deal with by eliminating the language used to talk about them.
That's no nefarious secret conspiracy. The President has repeatedly demonstrated the level of pettiness required to insist on this trick. Only when forced by substantial, sustained opinion does the President even acknowledge facts he'd rather avoid, let alone deal with them.