Do new wire runs in an existing ceiling need to be stapled down?
If you are not removing more drywall than needed for the new recessed lights, then no per 334.30(B)(1) (assuming you are using NM cable). There are similar clauses for other flexible cable assemblies (e.g. AC/MC).
However, if you are removing drywall as part of the remodel and have access to the studs, then you do need to secure and support according the remainder of 334.30 (or similar clauses for other cable types).

From NFPA-70:2014 (a.k.a. 2014 NEC):
334.30 Securing and Supporting Nonmetallic-sheathed cable shall be supported and secured by staples, cable ties, straps, hangers, or similar ﬁttings designed and installed so as not to damage the cable, at intervals not exceeding 1.4 m (4 1⁄2 ft) and within 300 mm (12 in.) of every outlet box. junction box, cabinet, or ﬁtting. Flat cables shall not be stapled on edge. Sections of cable protected from physical damage by raceway shall not be required to be secured within the raceway.
(A) Horizontal Runs Through Holes and Notches. In other than vertical runs, cables installed in accordance with 300.4 shall be considered to be supported and secured where such support does not exceed 1.4-m (4 1⁄2-ft) intervals and the nonmetallic-sheathed cable is securely fastened in place by an approved means within 300 mm (12 in.) of each box, cabinet, conduit body, or other nonmetallicsheathed cable termination.
FPN: See 314.17(C) for support where nonmetallic boxes are used
334.30(B) Unsupported Cables. Nonmetallic-sheathed cable shall be permitted to be unsupported where the cable:
(1) Is ﬁshed between access points through concealed spaces in ﬁnished buildings or structures and supporting is impracticable.
(2) Is not more than 1.4 m (4 1 ⁄2 ft) from the last point of cable support to the point of connection to a luminaire or other piece &gt; of electrical equipment and the cable and point of connection are &gt; within an accessible ceiling