I believe economics comes into the explanation of Ottoman success. Although the Byzantines had lost control over the tax income from land and trade (as I posted earlier), the Ottomans hadn't. The Ottomans were able to get the Jizya tax revenues and they also had no difficulty in getting their people to fight for the cause as ghazis, whereas the Byzantine nobles didn't pay tax and didn't fight, leading the state to rely on unreliable mercenaries. Also, the Ottomans could usually expand in one direction at a time, while the Byzantines had powerful enemies on all sides.
The Ottomans first appeared around 1299. The father of Osman I was Ertugrul; we don't know when he was born but he died in 1280. It is said that Ertugrul was the one who conquered the village of Sogut, which was called Thebasion until 1231. This was the origins of the Ottomans.
A PoD that prevents Byzantines from losing Western Anatolia probably needs to happen no later than 1290s. One can see that if this happens, it's likely Ottomans never emerge. One possibility was to make Alexios Anthropenos become Emperor in 1295 (earlier is better). It's difficult, because even here it may be too late. But he was a very good general and respected by Turks and Byzantines. Both of these served in his army.
It's not impossible that if he becomes emperor, we can see the Byzantines recruit a mixed army of Greeks and Turks and use this to gain control over some of the beyliks.
Success of the Ottomans was probably based on a mix of economic reasons, mentality of the society and organization. If the Byzantines want to rise again, it would be a good idea to reform the state significantly, to completely reorganize the way lands are given, who pays tax, and how the army is organized and paid. Who can do it? It would take the greatest emperor in their history, I think.