ExploderWizard
Hero
There are a number of options for non-combat resolution systems that prioritise player input and still involve dice rolling. The most generic way of describing those structures is probably this: the GM describes the situation/context, the player then describes what his/her PC is doing or saying (this may be first or third person, depending on individual preferences and table expectations), dice are then rolled, and the GM then narrates the consequences of the PC's action using the result of the roll(s) to establish the parameters of that narration (eg if the check is a failure, the GM's narration has to give some account, in the fiction, of the PC not getting what the player wanted him/her to get).
How much of what the player does or says influences the die roll?