Dial Indicator Lubrication

I would go for total immersion in white spirit, unless there are rubber parts in the DI, preferably in an ultrasonic tank and preferably dismantled. Simply adding oil to a dirty bearing will guarantee self destruction in a few years. When all is clean, re-assemble and lubricate with real oil, not muck like 3in1 or WD40.
Cliff Coggin Wealden Clock Repairs

The late, great George H Thomas defined 3 in 1 oil (by implication, not by actual name) a slow setting varnish and I absolutely concure with his opinion, he sugested that it was great for wheel barrows!
There a two types of suitable oil watch/clock oil and sewing machine oil both of which are
non-gumming oils. They should only applied in MINUTE qantities. They should be applied with an oiler which consists of a very fine wire. Unfortunatly the obvious thing to use is, a fine sewing needle is absoluely USELESS, If you dip a needle in oil in oil and examine the drop of oil you will observe that the drop is drawn up by surface tension so it is 0.5 - 1.00 mm away from the tip, and when you jiggle the tip arround to get the oil where you want it you will have applied much more than you wanted to. A thin wire of softish steel or even copper cut of with a rather blunt pair of side cutters will produce the required "spade" which will keep the oil drop at the end of the oiler so it transfers properly.
H S Walsh suitable oilers and oil: H012 "Trium" oiler set 3.53 H097 "Pioneer" watch oil 3.53 Post and Packing 4.95 Total 12.01
However for a one off job this is really too much to spend and if you could possibly find someone who is "into" horology and local to you I am sure they would do the oiling for free.
Regards Brian