The suffix -ham is very common in English placenames and personal names. In the UK the usual pronunciation is əm. In the US the pronunciation is həm or hæm. So in General British English:
Birmingham = ˈbɜːmɪŋəm
Rotherham = ˈrɒðərəm
Be careful of names where the element before the suffix ends in <p> or <t>. In these cases the combinations <ph> and <th> often do not represent the usual f and θ. So Clapham is pronounced ˈklæpəm not ˈklæfəm and Streatham is ˈstretəm not ˈstreθəm. Unfortunately, there are some exceptions. For example, Botham is pronounced ˈbəʊθəm.