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The name field supports three placeholders, which you can combine freely:
PlaceholderResolves to
[c]The parent composition’s name.
[l]The layer’s name. Only resolves when the new comp contains a single layer: one selected layer, or each comp when pre-composing separately.
[#]An auto-incrementing number. Use more hashes for zero-padding: [##] gives 01, [###] gives 001, and so on.

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How the index works

When the name contains [#], Pre-compose Pro looks through your project for existing comps matching the same pattern and continues from the highest number it finds, so [c]_precomp_[##] produces MyComp_precomp_01, then MyComp_precomp_02 the next time, with no duplicates.
The index placeholder must be at the end of the name. A default name like shot_[#]_element will be rejected.

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Prefix and suffix

In Settings you can define a prefix and suffix that are automatically added to every name, so if your naming convention marks every pre-comp the same way (a _precomp suffix, say) you never have to type it.

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Default name

Settings also has a Default field: the name the dialog starts with each time it opens. It supports the same placeholders, so once set, it means most pre-composes are just Enter and done.