I assume you want to weight the original "scores" not the ranks.
In another table you can use a VLOOKUP to get the score from each of the other tables, multiplied by the weight, e.g.
=VLOOKUP(A2,Table1,2,0)*0.6+VLOOKUP(A2,Table2,2,0)*0.2+VLOOKUP(A2,Table3,2,0)*0.2
Where A2 is the name and assuming that each table has the name in one column followed by the score in the next.
Now you can simply rank the "new" scores
Revised suggestion:
Given your comments I think you could still do the same as my initial suggestion except weight the ranks - I don't see how that won't be valid even if some of your original ranks are in descending order and some in ascending order........but if your rank is to the left of the name you can't use VLOOKUP so I would switch to SUMIF (although INDEX/MATCH would also work, so this formula in H2 copied down gives a combined rank assuming weights of 0.6/0.2/0.2 (adjust as required)
=SUMIF(C$2:C$4,G2,B$2:B$4)*0.6+SUMIF(C$7:C$9,G2,B$7:B$9)*0.2+SUMIF(C$12:C$14,G2,B$12:B$14)*0.2
and then you can get the overall rank with this formula in I2 copied down:
=RANK(H2,H$2:H$4,1)