Provided the VOR had a co-located DME, the best simple way was (and is) to take the DME distance as you fly due-south or due-north of the station (remembering to allow for the magnetic variation at the station). You know the longitude of the VOR, and simply apply the DME distance to the latitude. Try to do it when you are far enough away to minimise the DME slant-range error, but close enough to avoid a big error with the VOR bearing. I think about 30-90nm was considered a good compromise.
On the last generation of INSes, the position updating used the DME/DME as the most accurate.
The slant range error was minimised by entering
both the DME coordinates
and its MSL altitude. Some heavy manipulations, there...
As a matter of fact, on those old aircraft, one INS would provide navigational data to either the Captain (INS 1) or the First Officer (INS 2), INS 3 was used as an arbiter in case of vastly differing positions between 1 and 2.
Problem was that on autopilot/FD, the non-flying side would have a meaningless vertical bar after a while as the drifts were different.
Kept crews alert, though.