Step 1: The key fact is the ester versus amide split of local anaesthetics. Allergy almost always belongs to the ester family, so the management is to switch to an amide.
Step 2: A simple memory trick: amide agents carry two letter $i$ in their name (pr$i$loca$i$ne, l$i$doca$i$ne, bup$i$vaca$i$ne), whereas esters have only one. Applying this, prilocaine and bupivacaine are amides while cocaine, procaine and tetracaine are esters.
Step 3: Esters are cleaved by plasma esterases into para-aminobenzoic acid, the molecule responsible for hypersensitivity. Amides bypass this pathway through hepatic amidase metabolism, so they do not provoke the same allergic response.
Step 4: Therefore in an ester-allergic eye patient, an amide such as prilocaine is appropriate. Cocaine, procaine and tetracaine remain contraindicated.
\[\boxed{\text{Prilocaine (an amide local anaesthetic)}}\]