Name for the stick running through the small and narrow leaf of a big coconut leaf

The "spine" of both the leaf and each pinna of the leaf appears to be called the midrib.

Detail of coconut leaf [Adityamadhav83 via Wikimedia]

Cocos nucifera is a large palm, growing up to 30 m (98 ft) tall, with pinnate leaves 4–6 m (13–20 ft) long, and pinnae 60–90 cm long

[Wikipedia]

Pinnately veined leaves have one large central vein, called the midrib, which extends from the base of the blade to its tip.

[Robinson Library]

midrib
A large strengthened vein along the midline of a leaf.

[ODO]

Leaflet midrib

from uoregon.edu

Figure 10.-Coconut tree and its parts.

a, tree: 1, trunk (rakau); 2, base of trunk (tona); 3, roots (aka); 4, leaf (rou niu); 5, center keaves (tira).

b, leaf parts: 1, midrib (takai niu); 2, leaflet (mata rou niu); 3, leaflet midrib (tuaniu).

c, flower parts: 1,whole flower (karoro); 2, flower sheath (taume); 3, stalk of nut (pa karihi); 4, stalk of bunch (kauroro).

d, mature fruit: 1, outer skin (kiri taha); 2, husk (puru); 3, shell (ipu); 4, flesh (kaniu); 5, fluid (nia wai, plural); 6, nut stalk (pa karihi);.

e, growing nut (homo): 1, roots (aka); 2, leaf stipule (kaka); 3, leaf (rou homo); 4, central leaf (tira homo); 5, spongy interior (upu).

Older finds

I found this image at bioversityinternational.org:

enter image description here

but this one calls your specific part a midrib (rachis):

I think the term used in botany is rachis:

In plants, a rachis is the main axis of a compound structure. It can be the main stem of a compound leaf, such as in Acacia or ferns, or the main, flower-bearing portion of an inflorescence above a supporting peduncle.

Source:http://en.m.wikipedia.org/wiki/Rachis#In_botany