Nomenclature of lithiated dithianes
The exact nomenclature of alkyllithiums is an inorganic problem (cf. 2005 Red Book) and in there one finds as an example (IR-10.3.2, p 229; repeated in P-69.3, p 1043 of the 2013 Blue Book)

$\ce{LiMe}$ lithium methanide (compositional name)
$\ce{[LiMe]}$ methyllithium
$\ce{[(LiMe)4]}$ tetra-µ3-methyl-tetralithium
$\ce{(LiMe)_n}$ poly(methyllithium)

so, the name you choose will depend on what structural features you want to emphasise. Note that the Red Book does not prescribe &quot;preferred&quot; IUPAC names, and the Blue Book explicitly refuses to assign PINs (p 1036):

Neither preferred IUPAC names or preselected names (see P-12.2) for organometallic compounds involving the transition elements (including the Group 3 elements) and Groups 1 and 2 elements, excpet for 'ocene' compounds, are noted. This determination will await consideration by a task group on inorganic and/or organometallic nomenclature.

Obviously, from a practical point of view, the name &quot;methyllithium&quot; is more commonly used and more easily recognised. In line with this, we will therefore look for a name of the form &quot;_yllithium&quot;. The use of &quot;methyl&quot; instead of &quot;methanide&quot; is permissible because (p 1037, 2013 Blue Book)

Because of a long established tradition, names for substituent groups as defined in Section P-29 can also be used as ligand names.

and hence the question becomes: what is the preferred prefix for the following group?

This is a purely organic question, so can be answered with the 2013 Blue Book. The relevant rules are as follows:

P-29.2 [...] (2) The suffixes 'yl', 'ylidene', and 'ylidyne' are added to the name of the parent hydride with elision of the terminal letter 'e', if present, when followed immediately by the letter 'y'. The locants for the atoms of free valences are as low as is consistent with any established numbering of the parent hydride and, except for mononuclear parent hydrides or the suffix 'ylidyne', the locant '1' must be cited. This method is used to generate names of 'alkanyl-type substituent groups' that are simple substituent groups with free valences at positions other than '1'.
P-57.1.5.1 Preferred prefixes derived from cycloalkanes are of the 'cycloalkyl type' (see P-29.2); preferred prefixes derived from cyclic compounds other than cycloalkanes are all of the 'alkanyl type'.

The preferred IUPAC name of the parent hydride, according to Hantzsch–Widman nomenclature (P-22.2.2.1), is 1,3-dithiane. Correspondingly, the preferred prefix is 1,3-dithian-2-yl:

Finally, returning to the Red Book:

IR-2.2.3.2 Parentheses are used in names in the following ways:
(a) To enclose substituent group or ligand names in order to avoid ambiguity, for example if the substituent group or ligand names contain multiplicative prefixes, such as (dioxido) or (triphosphato), or if substitution patterns would otherwise not be unambiguously specified, or if the substituent group or ligand name contains numerical or letter descriptors [...]

we can conclude that a suitable name would be (1,3-dithian-2-yl)lithium.