Python 2, 405 bytes

Call w(e) to print the weight of element e:

def w(e):import re;print reduce(lambda c,(_,d):c+int(d)-1,re.findall(r"(\D+)(\d+)",re.match(".*%s\d"%e,"H2He4Li4Be3B3C2N3O3F4Ne2Na4Mg2Al4Si2P4S2Cl4Ar6K0Ca2Sc6Ti4V4Cr2Mn4Fe2Co4Ni1Cu6Zn2Ga6Ge4As3Se5Br2Kr5Rb2Sr4Y2Zr3Nb3Mo4Tc3Ru4Rh3Pd4Ag3Cd5In4Sn5Sb4Te7I0Xe5Cs3Ba5La3Ce2Pr2Nd4Pm2Sm6Eu3Gd6Tb3Dy5Ho3Er3Tm3Yb5Lu3Hf4Ta4W4Re3Os5Ir3Pt4Au3Hg5Tl4Pb4Bi3Po1At2Rn13Fr2Ra4Ac2Th6Pa0U8Np0Pu8Am0Cm5Bk1Cf5Es2Fm6").group()),0)

Tests:

w("O")
w("Sn")
w("H")
w("Fm")

Output:

16
119
1
257