When Robert’s Rebellion was over house Baratheon was the new royal family, but in a much weaker position than the Targaryens, whose authority ultimately came from the dragons they initially controlled. How could Robert and Jon Arryn have prevented the overlarge Lannister influence? Focusing on house Baratheon and the Stormlands or giving all the great houses some input into the matters of the state?

Basically, by playing dynastic politics better. To begin with, Stannis and Renly should have been raised to Princely status, to make it very clear that House Baratheon was here to stay. From a purely dynastic perspective, I think I would have made Stannis Prince of Summerhall as the new heir apparent title, since the name is associated with the destruction of House Targaryen and the glorious victory of the new King over his disloyal vassal, and given him Storm’s End to make sure that the adult brother has some real weight he can throw around. Put Ser Courtney Penrose in as castellan of Dragonstone, he’ll hold it until Renly can come of age. Might also want to bring in Estermont kin too, see if we can’t get some younger sons to form a cadet branch or something. 

Next, dynastic balance. Both Stannis and Renly should have been betrothed and married at earliest possible moments, to highest possible houses (Florents nowhere near good enough!), so that the Lannisters are primus among many pares. My first priority would have been to bring the Tyrells into the fold; they’re way too powerful and loyalist to be safely left out of it. So try for Mina or Janna Tyrell if their marriages hadn’t happened yet, failing that a future betrothal with Margaery is a good idea. Failing that, Malora Hightower or Desmera Redwyne would be good fallbacks. 

Now, it’s true that in 283, the ranks of unmarried daughters seem to have been a bit thin, but that’s where engaging newborn and/or hypothetical children can be quite handy. If the Baratheon monarchy is going to remain, it’s going to need to keep the Southron Alliance going – so make sure that the Starks, Arryns, Tullys, and Baratheons’ kids are all engaged to one another. 

Finally, begin building a king’s party at the court. Having Stannis and Renly on the Small Council is a great idea, but there’s no reason why you couldn’t have given Hoster Tully a seat and had Edmure as a page/squire, maybe bring the Blackfish into the Kingsguard. Reach out to the more powerful lesser Houses, give them some patronage too – the Royces/Waynwoods/Hunters/Redforts, Penroses/Carons/Swanns/Estermonts, Whents/Blackwoods or Brackens, maybe some Manderlys?