According to the current version of Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book), secondary and tertiary amines can be systematically named in various ways similar to the methods for primary amines:
P-62.2.2 Secondary and tertiary amines
P-220.127.116.11 Symmetrical and unsymmetrical secondary and tertiary amines are named only by the same methods described in P-18.104.22.168.
(1) substitutively using the retained name ‘aniline’ or the suffix ‘amine’ and the name of a parent hydride with further N-substitution;
(2) substitutively, by prefixing, in alphabetical order when required, the name(s) of the substituent group(s) R, R' or R" to the parent hydride name ‘azane’. In order to avoid ambiguity, the second prefix in a secondary amine, and the second and the third prefixes in a tertiary amines must be enclosed in parentheses when these prefixes denote simple substituents.
(3) substitutively, by prefixing, in alphabetical order when required, the name(s) of the substituent group(s) R, R' or R" to the parent hydride name ‘amine’. In order to avoid ambiguity, the second prefix in a secondary amine, and the second and the third prefixes in a tertiary amines must be enclosed in parentheses when these prefixes denote simple substituents.
Method (1) generates preferred IUPAC names. Functional parent names like diethylamine and triethylamine are deprecated. The prefixes in names of such secondary and tertiary amines formed by method (2) are set off by parentheses to distinguish them from these deprecated names.
Therefore, the preferred IUPAC name for the tertiary amine given in the question is generated according to method (1) using the suffix ‘amine’, the name of a parent hydride, and further N-substitution. (Other methods may be used in general nomenclature.)
Concerning the choice of the principal chain, the rules for amines state:
P-22.214.171.124 Selection of the principal chain or senior ring system in secondary and tertiary amines.
Names of amines formed substitutively by using the retained name aniline or the suffix ‘amine’ are based on a principal chain and a senior ring system (see P-44.1). When a choice for parent hydride is possible between a ring and a chain, the ring is preferred. In names using ‘amine’ as a parent hydride, substituent groups expressed as prefixes are cited in alphanumerical order; the prefix(es) immediately preceding the term ‘amine’ is (are) enclosed in parentheses.
Thus, the rules essentially refer to the usual seniority order for parent structures that is stipulated in P-44.1 and the following subsections. The relevant criteria for the compound given in the question are
P-44.1.1 The senior parent structure has the maximum number of substituents corresponding to the principal characteristic group (suffix) or senior parent hydride in accord with the seniority of classes (…) and the seniority of suffixes (…).
P-44.3.2 The principal chain has the greater number of skeletal atoms (…)
In case of the compound given in the question, both methane (methylamine) and ethane (ethylamine) are parent structures that have the maximum number of substitutents corrsponding to the principal characteristic group (i.e. the suffix ‘amine’) since the compound contains only one amine group. The principal chain for the compound given in the question is ethane rather than methane since it has the greater number of skeletal atoms (i.e. the longest chain).
In accordance with the above-mentioned method (1), using the name of the parent hydride ‘ethane’ and the suffix ‘amine’ with the appropriate locant yields ‘ethane-1-amine’. The chloro substitutent is placed at atom ‘2’ of the parent hydride ‘ethane’, which leads to
‘2-chloroethan-1-amine’. Further N-substitution with the substituents ‘methyl’ and ‘2-chloroethyl’ results in the preferred IUPAC name
Note that the locant ‘1’ is omitted in ‘ethanamine’; however all locants must be cited in ‘2-chloroethan-1-amine’ and ‘2-chloro-N-(2-chloroethyl)-N-methylethan-1-amine’.
The alternative name bis(2-chloroethyl)(methyl)amine can be generated according to method (3). This is not the preferred IUPAC name, but it is a correct systematic name that may be used in general nomenclature.
Note that, according to method (3), the parent hydride in this case is ‘amine’ (i.e. neither ethane nor methane). The names of the substituent group(s) R, R' or R" are added as prefixes in alphabetical order, for example: ‘ethyl(methyl)(propyl)amine’. The multiplicative prefixes ‘di’ or ‘tri’ are added as usual, for example: ‘diethyl(methyl)amine’. However, the multiplicative prefixes ‘bis’ and ‘tris’ are used for substituted substituents (such as ‘2-chloroethyl’) and other compound or complex entities, which results in the name bis(2-chloroethyl)(methyl)amine for the compound given in the question.
Note the correct use of parentheses according to Subsections P-126.96.36.199 (3) and P-188.8.131.52.
The name N,N-di(2-chloroethyl)methanamine, which has been proposed in the question, is simply wrong. It violates multiple nomenclature rules.