I'd say 17. Since Gimli isn't quite as mutant-heavy as D&D Dwarves are, I figure he'd be about 170 pounds in chain armor. At 17 Str, Gimli would be only medium encumbrance, so Aragorn only needed his DC 10 Climb check with a -3 Armor Check Penalty (assuming his armor didn't make that worse already) to do the rope trick. If Aragorn put any ranks into Climb (he probably did), he doesn't need to be that lucky. If that rope had a knot in it, the check was DC 5 and almost a sure thing!
The tossing is harder to judge, since no rules exist in D&D to decide such issues. At Str 17, Aragorn can easily pick Gimli up and still move, so tossing him a short distance shouldn't be TOO big a deal, say a DC 17 Strength check. That means Aragorn got a good roll.
If you figure Gimli couldn't encumber Aragorn at all to do such things, then Aragorn would need to be Str 22. This seems a little high, since he would have fared better in his wrestling against Lurtz in the first movie if it were that high!