The President cannot initiate Constitutional amendments or changes through the popular initiative under Article 257 of the Constitution. -  KESC 8 (KLR)
The bulk of legislative proposals considered and enacted by Parliament are made by the national executive. It is the rare private member's Bill that is enacted by Parliament and assented to by the President. This is to say that for the most part, the President can send a legislative proposal to Parliament and Parliament, in its wisdom as the representation of the legislative will of the people, can enact it or reject it. Therefore, the President can initiate an amendment to the Constitution of Kenya. What he can't do, and the High Court, Court of Appeal and Supreme Court were adamant on this point, is to initiate Constitutional amendments by popular initiative.
It is important to remember one other thing: an amendment to the Constitution by popular initiative does not necessarily mean that a referendum must be conducted on the amendment. A referendum must be conducted if the amendment affects any of the issues found in Article 255. The question of whether or not a referendum is warranted is independent of whether the amendment was by parliamentary or by popular initiative.
As to the question of the "office of the leader of the official opposition", a referendum is definitely warranted. Going by how the Azimio One Kenya Alliance described the functions of the proposed office, it is clear that the proposed amendment falls squarely within the ambit of Article 255 ("the functions of Parliament") and on that basis alone, whether the proposals is a parliamentary one or a popular one, a referendum is necessary for the amendment to be adopted by the people.
Regardless of who brings the amendment process to life, we are being manipulated onto thinking that the amendment is not a bad one. The popular discourse regarding the proposal is that it is a new form of handshake between the new administration and the putative opposition, and it is almost always framed in those terms. The Minority Party in Parliament is not treated as part of Parliament but an Opposition Party in the Westminster style of parliamentary government. It is not. It hasn't been since 2013. But you wouldn't know it going by how the 11th and 12th Parliaments behaved with Parliamentary Parties taking their political differences to such dysfunctional depths that Parliament never effectively played its oversight role over the national executive. In short, Parliament shirked its role and the Jubilee administration got away with all manner of dodgy things.
If we allow the office of the leader of the official opposition to be established along the lines proposed by Azimio, it is almost certain to be the last nail in the coffin of parliamentary oversight and we can kiss whatever restraints there were on the national executive goodbye. First, oversight is not an individual endeavour but an institutional one. A single "office" is incapable of wielding the kind of collective power that Parliament, and parliamentary committees, can. Secondly, and more crucially, it absolves Parliament from the responsibility of lax oversight; parliamentarians can pass the buck and say, "Hey, the Leader of the Official Opposition didn't do this, that or the other" and they can play footsie with the executive to their hearts' content. If the office is established, it is the people who will come out the poorer for it.
Mr. Odinga, and his acolytes including Prof. Makau Mutua, have had a good innings at the hustings but they are more likely today to advance political ideas that are twenty years out of date, like the cult of personality they insist on building around Mr. Odinga. Mr. Odinga's ODM is no longer capable of fresh ideas; listening to the younger members of the party leadership (like Sen. Sifuna and Hon. Ongili), one harkens back to the Kanu of the late 1970s with the likes of Kihika Kimani and Kariuki Chotara springing readily to mind who were determined to sustain a cult of personality around Jomo Kenyatta even though by 1975 it was clear he was over the hill and clinging on to power by the skin of his teeth. It is time to put Mr. Odinga, Prof. Makau and their generation of the Old Guard of the Second Liberation out to pasture, and bury moribund and retrogressive ideas like the Office of the Leader of the Official Opposition in the deepest pit we can find.