This depends a bit on the version of the LGPL.
The LGPLv3 is not a standalone license, but an Additional Permission that can be applied to the GPLv3. Thus, your licensing documentation should include both the GPLv3 and the LGPLv3. This can be achieved by putting each into a separate file, or by putting both into the same file.
The LGPLv2 is a standalone license. If you use it, you should place it into any reasonably named file and not include the GPL alongside it.
The name of the file that contains the license terms is not so important, as long as it is crystal clear to recipients of the software which terms apply. For example, your README file should include a statement to the effect “This software is covered by that license”, and where to find the full terms.
Currently, I am preparing software for publication where this license statement is contained in the README and LICENSE files, all source files list the SPDX license identifier, but the full license terms are included in the bundled documentation. The result is that Github can't detect the license in its GUI, but this makes it easier to deal with complex licensing situations.