Damage is a natural place to insert die rolls, damage is almost always a bunch of die rolls already. x d[p] damage could be pretty standard, burning your focus or PP for additional d[p] the same way upcasting works for casters.

Duration of an effect maybe not so straightforward. I could see it being a plausible substitute for Concentration if the duration was kept short as in [p] rounds instead of minutes, especially for powers that you would be using in combat (specifically to avoid the problems you see in Tenser's Transformation, Stoneskin and the like).

Range is another area that could be possible, if a bit weird. You probably don't want to do anything like straight [p] x5 feet as that would be too random, perhaps a base range plus [p] extra or make a little table that splits ranges into bands (eg 1-3=30', 4-6=60', 7-8=90', 9-10=120')

What other factors would there be to fiddle with... number of targets affected? Could use [p] in the same way as Range is determined, base + roll or factor into a table. 'Casting' time? Probably better suited to resource expenditure and features. Perhaps using [p] to affect attacks/checks/saves, adding to allies or taking from enemies in a way similar to Bless/Bane? Probably better suited as a feature again but would tie in nicely.

You could cheat and take a look at all the ways Monks and Battlemasters incorporate their MA and superiority die into things.