What is significant about the half-sum of positive roots?

I don't think there is a one-line answer to this question, since it depends a lot on the direction from which you approach semi-simple Lie theory. For one thing, it's probably best at first to emphasize just integral weights, among which the dominant ones parametrize irreducible finite dimensional representations. Here the weight usually denoted $\rho$ plays a ubiquitous role in the classical Weyl theory, but that too can be developed in a number of different ways. (There was some early experimentation with the notation; the alternative symbol $\delta$ also had widespread use before the Bourbaki preference for $\rho$ started to take over in 1968.)

While it's important in proofs of the Weyl character formula to view $\rho$ as the half-sum of positive roots (given a fixed positive or simple system), it's also essential to identify it with the sum of fundamental dominant weights for many purposes. In this guise it's the smallest regular dominant weight, fixed by no element of the Weyl group except the identity. When passing from integral weights to line bundles on an associated flag variety $G/B$ (with $B$ a Borel subgroup associated to positive roots relative to a fixed maximal torus which it contains), the weight $\rho$ has the distinction of defining an ample line bundle. This property is crucial in geometric approaches to Weyl's formula, as well as in spin-offs in prime characteristic due to Andersen and others.

Ultimately the importance of the weight $\rho$ is probably appreciated best in the setting of representation theory, where the finite dimensional theory is enriched by treatment of highest weight modules in more generality and the shift by $\rho$ is again ubiquitous. By the way, the convenient "dot" notation $w \cdot \lambda := w(\lambda +\rho) - \rho$ is apparently due to Robert Moody. In the earlier literature the more awkward full notation appears, or else is replaced in the Paris notation by a hidden $\rho$-shift.

None of what I've said is a complete answer to the question asked, but in any case it's more than a matter of "convenience" to emphasize $\rho$.

From the point of view of geometry, the crucial fact about $\rho$ is that the corresponding line bundle on the flag manifold is (upt to a sign) a (the) square-root of the canonical bundle (top exterior power of $T^*_B G/B \simeq b_- $ is the sum of the negative roots). This is of course equivalent to Alain Valette's description in terms of the modular character of the Borel. In other words its sections in the real world are half-densities (things for which we can define the $L^2$ inner product).

It is a universal fact about passage from the classical world to the quantum world (in particular the geometric construction of representations) involves a shift by the square root of the canonical bundle. There are many ways to explain or motivate this. For example if we seek unitary representations we need to be able to define an $L^2$ inner product, which means considering not sections of the bundle we might have expected but sections times half-densities (again this is Alain's answer restated). From the point of view of rings of differential operators, the adjoint of a differential operator acting on functions (or on sections of a bundle $L$) is invariantly not another diffop (on $L$) but a differential operator acting on volume forms (or on sections of $L$ tensor the canonical bundle) --- so the self dual twist of differential operators is by half-forms, ie $\rho$-shifted. (Put another way, Serre duality is a reflection centered at half-forms!)

My favorite explanation is in Beilinson-Bernstein's Proof of Jantzen Conjectures and doesn't involve self-adjointness or unitarity: it's a consistency condition for deformation quantization of symbols (functions on the cotangent bundle): if you want this deformation quantization to be correctly normalized to order two (this is not the right question to go into that) you find you need to look at differential operators twisted by half-forms, not functions. On the flag variety this means a $\rho$-shift, and from the D-module POV on representation theory this is one fundamental place where that shift is forced on you, independent of thinking of inner products. This is in particular one way to see why it comes up in the Weyl character formula, through the geometric proof via Atiyah-Bott or via the BGG resolution, both of which involve the geometry of the flag variety.

This is actually a fairly deep question. Your suspicion that there may be multiple answers is correct, but there might be some surprising connections between seemingly unrelated answers. Let me give one possible thread of explanation. The underlying principle is that the appearance of $\rho$ and the "dot" action $w\cdot\lambda=w(\lambda+\rho)-\rho$ in representation theory is closely related to the geometry of the flag variety.

One of the first places one meets $\rho$ (and the dot action) is in the Weyl character formula. A theorem of Kostant shows that the formula can be written as the ratio of two Lie algebra cohomology Euler characteristics. From this perspective, the appearance $w \cdot \lambda$ and $w\cdot0$ in the WCF is ultimately explained by the fact that these are the weights appearing in the weight space decomposition of the relevant Lie algebra cohomology modules, namely $H^*(\mathfrak n, V^\lambda)$ and $H^\ast(\mathfrak n, V^0)$, where $\mathfrak n = \bigoplus_{\alpha>0} \mathfrak g_\alpha$ and $V^\mu$ denotes the irrep of highest weight $\mu$.

We can rephrase this in geometric terms by invoking the "geometric analogue" of Kostant's theorem, i.e. the Borel–Weil–Bott theorem. Kostant's description of the Lie algebra cohomology of $\mathfrak n = \mathfrak g /\mathfrak b^-$ with coefficients in an irrep translates into a representation-theoretic description of the sheaf cohomology of certain line bundles $L_\lambda$ (constructed using integral weights $\lambda$) over the flag variety $G/B^-$ of $\mathfrak g$. Consequently, the dot action shows up in this description, and this time it's accompanied by a shift in degree. This in turn can be explained by Serre duality; the key fact is that canonical bundle of $G/B^-$ turns out to be $L_{-2\rho}$.

So, in some sense, the appearance of $\rho$ and the dot action in the WCF can be thought of as a manifestation of Serre duality.

[N.B. This is a condensed version of my lengthy original answer. The old version can be found in the edit history.]