I would not use a variac for the permanent operation of the station: the risk of a misplaced knob is too hight.
There are two cases:
1) the Weller station is built in Europe and sent to U.S. : it can have a dual-primary transformer, rated at 50 Hz (thus it will have no problems at 60 Hz), and by just moving a couple of jumpers it can be converted to 230 V or 115 V operation, or a bigger transformer with primary taps at 115 and 230 V.
Maybe searching the station's service manual will give a clue.
2) The Weller station is built in the USA, and it has a (slightly cheaper to manufacture) single primary transformer, and they saved every penny by designing the transformer for 60 Hz only (maybe 50 grams of iron saved...). In this case the transformer will be hotter at 50 Hz, but the mean power delivered during operation will be less than 100% (due to heat cycling to the soldering iron), and this will not be of any harm.
I looked inside one of my Weller stations: it has a transformer with dual windings , but I don't know if this is the rule.
In the years I bought a lot of test instruments from the USA, when they were too expensive here in Italy, and most of the times I disassembled the power transformer and had it rewound for 230 V, with a slightly smaller wire and maintaining the original iron, winding former and secondary winding intact (Making a new transformer will have resulted in some difficulties in mechanical compatibility)
They are still operating after some decades and, yes, the transformers are a little warm, but cooler than most modern equipment.