Most vanity drains use 1 1/4” pipe. Go to someplace like HD and get a 1 1/2” kitchen pop up assembly. These are used in kitchen sinks. The slightly larger diameter collar might cover the elongated opening. Use clear silicone under the collar flange to ensure a leak free seal. The tailpiece will be 1 1/2”, but they make reducers to 1 1/4”.

If you’re comfortable working around plumbing this is the way to go.  IMHO. 
 

Not knowing the exact size of your sink opening, try a 1 1/4” pop up assembly first. If that covers everything, go with that.

Is there a separate piece screwed into that drain? Show pic of underside.  I'm thinking a conventional house drain might fit and the lip would cover and seal around the old drain. Hard to tell from pics

Linda S

While fixing the leaking drain I complicated the issue. The sink was just too brittle and I made it worst. Anyone knows any resource for either replace the whole sink and counter top or somehow fix the whole so the drain can fit. 

Attached are two photos: