A padeye is a device often found on boats that a line runs through, or provides an attachment point. It is a kind of fairlead

Fairlead

A fairlead is a device to guide a line, rope or cable around an object, out of the way or to stop it from moving laterally. Typically a fairlead will be a ring or hook. The fairlead may be a separate piece of hardware, or it could be a hole in the structure....

and often is bolted or welded to the deck or hull of a boat.

It is also used in oil and gas projects to assist in the purpose of lifting.

Detail

It's made of steel plate with radius at oneside. lifting is done with the help of D-shackle or sling,which fits into the hole of padeye. there may be one or more circular plates(cheek plates) welded around the hole.

Designing

Following check should be done for the designing of padeyes
and keep the stress less than the allowable stresses

At the hole:

1. Bearing stress

2. Shear stress

3. Tensile stress

4. Hertz Bearing stress

At the base

1. Shear stress

2. Tensile stress

3. Bending stress

4. Combined bending stress and tensile stress

5. Von-Mises stress
A spreadsheet is more often used for this purpose.