Yeah — there is a name for this. A couple, actually, depending on how the sound is removed:
1. Aphaeresis
This is the term for dropping the first sound or syllable of a word.
Examples:
’cause (from because)
’em (from them)
ol’ (from old)
tis (from it is)
2. Apocope
This is the term for dropping the last sound or syllable of a word.
Examples:
fam (from family)
bio (from biology)
tho (from though)
maths → math (in American English)
3. Clipping (broader, more casual term)
This is the general linguistic term for shortening a word by removing any part of it (front, back, or middle).
Aphaeresis and apocope are just the more technical sub-types.
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If you want the most precise answer for “ol’,” that’s aphaeresis.