2. The name ua for the staff may be a pun, because ûa means not only 'rain' but also 'vein, artery, tendon':

"In the crude anatomical knowledge of these races it is easy to see what ua [ûa] really is, the cordlike bodies in the flesh which appear under the skin. Thus vein and tendon are the same thing and one word describes them." (Churchill 2)

The double triple lines in the ua glyphs may refer to the ua staff (pars pro toto).

The arched 'eyebrows' on the head of the ua staff may have led to the name ua for the glyph type (because ûa = 'vein, artery, tendon').

And maybe ua staffs were involved in ceremonies for inducing rain (ûa). One of the meanings of ua is 'cause, reason why something happens ':

"It was 4 August 1968, and it was the feast day of Saint Dominic, patron of Santo Domingo Pueblo, southwest of Santa Fe. At one end of the hot, dusty plaza, a Dominican priest watched nervously as several hundred dancers arranged in two long rows pounded the earth with their moccasined feet as a mighty, collective prayer for rain, accompanied by the powerful baritone singing of a chorus and the beat of drums.

As my family and I viewed this, the largest and in some ways the most impressive Native American public ceremony, a tiny cloud over the Jémez Mountains to the northwest got larger and larger, eventually filling up the sky; at last the storm broke, and the sky was crisscrossed by lightning and the pueblo resounded with peals of rolling thunder." (Coe)