There's a plausible theory that James suffered from either intermittent porphyria or Lesch-Nyhan syndrome, but it was mild enough that the mental symptoms weren't debilitating. Give him a more severe case, so he suffers obvious fits of madness before the English succession is firmly settled in his favor.
The candidates for the succession are then Edward Seymour, Anne Stanley, or Arabella Stuart.
In Seymour's favor, he's an adult man, and he has an heir and a spare. Working against him, his family has a rather unsavory reputation at this point, and the legitimacy of his birth is dubious at best: his parents married illegally, without documentation, and the priest is missing and the only witness to the marriage was conveniently dead when the marriage was discovered. The Privy Council at the time declared the marriage void, Seymour a bastard, and stripped Seymour's father of his lands and titles for the crime of "seducing a virgin of the blood royal". IOTL, James rehabilitated the Seymours and declared the marriage valid, but that was quite a bit after our POD.
Arabella was a serious contender for the succession in the 1580s, but she'd been firmly shelved in favor of James by the time Elizabeth died. Her claim over James relies on some legal hair-splitting (either applying the loosely-enforced principle that only an English subject can inherit English titles, or parsing the marriage treaty between James IV and Margaret Tudor as being a morganatic marriage for the purposes of the English succession), and she was an unmarried woman with no clear heir herself.
Anne Stanley had the best legal claim on paper, but was never seriously considered to be in the mix for the succession, mainly because she (like her father and grandmother before her) went out of her way to not appear to be advancing her claim. She was also an unmarried woman, but she's five years younger than Arabella (23 vs. 28), so there's more time for her to marry and produce heirs, and she has two younger sisters, one of whom was already married to the Earl of Huntingdon, so there's a clear succession after her if she were to fail to produce an hear. The other major handicap to her as a candidate is that she (like her father) was conspicuously quiet about religion, leading many to suspect that they were either Catholics or Puritans.
If James were disqualified by obvious fits of madness, I think there's a chance the Cecils might settle on Anne in preference to Arabella or Seymour. The more so if Seymour tries (as he may have tried IOTL) to strengthen his claim by marrying his eldest son to Arabella, which could backfire badly if he gets caught because marrying someone that close to the succession is illegal without the approval of the council, and is tantamount to treason (IOTL, Arabella married Seymour's second son William during James's reign; she was arrested and died in the tower, while William successfully fled to the Netherlands and was eventually forgiven by James).