Joe Brady.
He has been making some serious waves in the NFL as the Bills' OC, and was the genius behind Orgeron's national championship, fully unlocking Joe Burrow's abilities.
He's very much loved in Baton Rouge, and is a young coach who seems to have a very high ceiling. He's also very well-liked by almost all of the offensive players, with the few dissenters being the glory hounds who don't like how he has a "share the wealth" offense, where he tries to get everyone involved.
Brady's success as a HC would depend on his getting a good defensive coordinator, much like how Marcus Freeman's success was far more present after finally getting a real OC, but LSU has deep pockets, and they would definitely spring the money to get him a good one.