lime, Lyme

Lime or lime means the green citrus fruit--
He wants a wedge of lime to squeeze on his salad.
Lime may be used to describe something that looks like a lime--
He wants to paint the trim lime green.

Lyme ( usually capitalized, rhymes with lime ) is the name of a place; the disease borne by ticks was named for it--
The doctor thinks he may have Lyme disease.
Make sure to check for ticks; they can carry Lyme disease.

Now that you know that, you can say--
"It's called Lyme disease after a town in Connecticut; eating a lime doesn't have anything to do with it."