How to name molecules with a hydroxyl group in a side chain?
The proposed name 2-(hydroxymethyl)propane-1,3-diol is correct.



The current version of Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book) includes three different methods for systematic names of alcohols. The preferred names are formed using the suffix ‘ol’ and (if necessary) the prefix ‘hydroxy’.


  P-63.1.2 Systematic names of alcohols, phenols, enols, and ynols
  
  Hydroxy compounds are named in three ways:
  
  (1) substitutively, using the suffix ‘ol’ and the prefix ‘hydroxy’. The presence of several ‘ol’ characteristic groups is denoted by the numerical multiplying prefixes ‘di’, ‘tri’, etc.; the final letter ‘a’ in a multiplying prefix is elided before the suffix ‘ol’. Rule P-44 is applied when a principal chain or a senior ring system must be chosen. When there is a choice for numbering, the starting point and the direction of numbering of a compound are chosen so as to give lowest locants to the ‘ol’ suffixes;
  
  (2) by functional class nomenclature and the class term ‘alcohol’;
  
  (3) as assemblies of identical units by multiplicative nomenclature when the conditions for its use are fulfilled (see P-51.3)
  
  Method (1) generates preferred IUPAC names. Names as assemblies of identical units, method (3), are preferred to those that are formed by simple substitution (P-51.1.5).


The first step is the selection of the principal chain, which will serve as the parent structure. The relevant general criteria for the choice of a principal chain in such compounds are:


greater number of groups that are expressed as suffix
longest chain
lower locants for groups that are expressed as suffix



  P-44.1 SENIORITY ORDER FOR PARENT STRUCTURES
  
  When there is a choice, the senior parent structure is chosen by applying the following criteria, in order, until a decision is reached. These criteria must always be applied before those applicable to rings and ring systems (see P-44.2) and to chains (see P-44.3). Then criteria applicable to both chains and rings or ring systems given in P-44.4 are considered.
  
  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 (P-41) and the seniority of suffixes (P-43).
  
  (…)
  
  P-44.3.2 The principal chain has the greater number of skeletal atoms [criterion (b) in P-44.3].
  
  (…)
  
  P-44.4.1 If the criteria of P-44.1 through P-44.3, where applicable, do not effect a choice of a senior parent structure, the following criteria are applied successively until there are no alternatives remaining. These criteria are illustrated in P-44.4.1.1 through P-44.4.1.12.
  
  The senior ring, ring system, or principal chain:
  
  (…)
  
  (h) has the lower locant for an attached group expressed as a suffix (P-44.4.1.8);
  
  (…)


For the compound that is given in the question, the principal characteristic group is $\ce{-OH}$, which is expressed as suffix ‘ol’. Whichever way you look at it, the principal chain can only have two of these groups. Therefore, the principal chain is a propane; the parent structure including the substituents corresponding to the principal characteristic group is propane-1,3-diol.



The compound that is given in the question actually has a branched chain structure. Therefore, the next step is the identification and naming of side chains, which will be treated as prefixes. In this case, the resulting name is 2-methylpropane-1,3-diol.



The third $\ce{-OH}$ group cannot be expressed as suffix ‘ol’ since it is not situated on the principal chain. Therefore, it has to be indicated by the prefix ‘hydroxy’.


  P-63.1.4 Substitutive nomenclature, prefix mode
  
  Hydroxy groups are indicated by the prefix ‘hydroxy’ when:
  
  (1) a group having priority for citation as the principal characteristic group is present; or
  
  (2) a hydroxy group cannot be denoted by a suffix.


In this case, the hydroxy group is placed on the methyl side chain, which results in the compound substituent group ‘hydroxymethyl’.


  P-29.4 COMPOUND SUBSTITUENT GROUPS
  
  P-29.4.1 Compound substituted substituent groups
  
  A compound substituted substituent group is formed by substituting one or more simple substituents into another simple substituent that is considered as the principal chain. The choice of the principal chain is fully discussed in Section P-44.3. The first criterion to be applied is that the principal chain is the longest chain; it will be applied in the following examples of acyclic compound substituent groups.
  
  (…)


Parentheses are used around such compound prefixes in order to avoid ambiguity.


  P-16.5.1 Parentheses (also called curves or round brackets)
  
  P-16.5.1.1 Parentheses are used around compound (see P-29.1.2) and complex (see P-29.1.3) prefixes; after the multiplicative prefixes ‘bis’, ‘tris’, etc.; to enclose a multiplied parent compound consisting of a parent hydride and a substituent suffix or a multiplied substituent prefix even though preceded by a basic numerical prefix, such as ‘di’, ‘tri’, etc.


Therefore, the complete systematic name for the compound that is given in the question is 2-(hydroxymethyl)propane-1,3-diol.