Lord Paramount Lyonel Baratheon
Though they were nearly rendered extinct by the War of the Five Kings, House Baratheon survived thanks to the efforts of Davos Seaworth. While Stannis Baratheon fought valiantly both against the tyrannical House Bolton and the monstrous Others, he eventually sacrificed his own daughter, Shireen, as a last ditch effort to push back the Walkers. It was unsuccessful, and Stannis died a broken man. However, one of King Robert's bastards had joined the Brotherhood Without Banners, and through his heroism was legitimized as Gendry Baratheon at the conclusion of the War for the Dawn. However, Ser Gendry was averse and ill suited to rule Storm's End, and freely abdicated to Edric Storm upon his return from hiding in Essos and retired to a life of smithing at the Crossroads Inn along the Kingsroad.
Edric's reign as Lord Paramount of the Stormlands was rather plain and uneventful. For much of the first half of the 300s AC, House Baratheon was content to stay in its homeland, replenish its numbers, and reassert itself over the region. That was, until Davos Baratheon took over as the head of house.
Where his father had strengthened ties with their immediate vassals, Davos came to a conclusion the rest of Westeros would take decades to come to themselves; Queen Daenerys wasn't going to name an heir, and when she inevitably came to same end as all humans do, the power vacuum left in her wake would be extraordinary, and to come out on top, the Baratheons would need the support of the other Great Houses.
To this end, Lord Davos made a number of deals with the wider Realm; his elder son and heir sent to Seagard to serve under the Lord Paramount of the Trident, his second son squired at the Eyrie, and his only daughter betrothed to the now Warden of the North. He even sought a seat on the Small Council for himself and spent much of his time in the Capital, but was never appointed, to his great dismay.
The biggest obstacle to Davos' plan came when his firstborn son was killed at the age of six and ten. This put the Baratheons' potential alliance with the Mallisters in jeopardy, and Davos moved swiftly to ensure such a thing was still viable. While he still had a son to spare, however, said son was hundreds of miles away in the Vale, and he was decidedly against his father's wish for a prearranged marriage.
Lyonel Baratheon's time as heir to Storm's End did not start particularly well. Young and in love, he went against convention and married Rylene Redfort in a private ceremony in the Eyrie without his father, or Lord Mallister's, knowledge. To say Lords Davos and Artys were displeased when they arrived in the Vale with Lyonel's so called betrothed only to find a newly married man with a pregnant wife would be an understatement.
(Davos dying here, Stepstones Rebellion, Lyonel's appointment to and service on the Small Council.)
The Baratheons find themselves at a crossroads of opportunity. Where they were once rendered all but extinct by the War of the Five Kings, with Daenerys' death looming on the horizon, they stand to regain their place upon the Iron Throne of Westeros. Dominion over the Seven Kingdoms is not so easily gained, however. While the Lord Paramount of the Stormlands has made no secret of his wish to call a Great Council, his former peers on the Small Council were less than receptive of the idea. For now, the Baratheons bide their time in Storm's End.