Bush at the 2008 RNC Convention
The game’s on to guess whether George W Bush will be speaking at the RNC Convention. His absence would likely have to be more problematic than his presence — in 1984, the Democrats rushed through Jimmy Carter for that reason. But then, he was not sitting president.
I cannot think of precedents for an absent sitting president not being spotlighted at their convention due to high unpopularity. Woodrow Wilson was infirm in 1920 (and indeed, wife Edith pretty much the acting president), and unhighly unpopular nation-wide, but the Democrats churned out some well greetings or others. I think Lyndon Johnson’s name was avoided at the 1968 Democratic Convention. (If Nixon had staved off impeachment, he might have fallen into this trap, as it were President Ford made minor headlines by passing out his name in his speech.) After those example, I’ve got nothing.