Thanks to everybody for criticism and constructive comments.
I'd like to summarize the objections to the use of "hu" as a gender- neutral pronoun.
I'll respond point by point, first regarding the oral "hu" [hju:], then
the written "hu". In fact, it was my fault to ask only
"how does it sound to you?" In the first place, I should
have asked "how does it look?" A gender-neutral pronoun
seems to be a more urgent need in written language where a
word's social and ethical effects cannot be supported or
softened by an intonation, gesture, etc.

Oral "hu".
1. "Hu" is omophonic with the name Hugh.
--I don't think that there will be much confusion about it: Hugh
occurs only once in 1666 males; the name's popularity rank in
the U.S. is #254. Even more popular proper names are safe
from possible confusion with their common name doubles. Nobody confuses
Nick with a small cut, or Rick with a stack of hay.

2. "Hu" [hju:] may be confused with "who" [hu:].
--The phonetic distinction between t! hese words belongs to
the differential structures. [j] is a separate phoneme
that serves to distinguish lexical units. Cf. feud [fju:d]
and food [fu:d]; nuke [nju:k] and nook [nu:k]; hew [hju:]
and who [hu:].

3. "Hu" can be pronounced or heard as "you."
--There are many dialects and manners of
pronunciation, however nobody suggests to ban the word "air"
because it may be confused with "hair" or to drop the word
"hear" because it may sound similar to "ear" (even as
they are collocutives used together in many contexts).

Written "hu".

4. "Hu" looks like a Chinese word.
--Yes, but no more so than "van," a clipping of an
exotic word "caravan" (looking like "Chinese" doesn't prevent "van" from
being one of the most usable English words, 48 mln. in Google).
"Hu," as a syllable, is common to English: "huge, humor, human, humility..."
Both the pronunciation [hju:] and the spelling "hu" are quite ordinary, there is
nothing exotic about them.

5. The apostrophe in the possessive form "hu's". What is
contracted here?
--There is no contraction here. The apostrophe is
a sign of the possessive case, like in "John's, author's,
book's," etc. It would be even better to drop the apostrophe
if we could have "hus" pronounced [hju:z], not [has], as in "bus";
and "hum" pronounced as [hju:m].

Examples:

Hu that has ears to hear, let hum hear.

It's the vice-president's job to support the president and take
hus place when hu is away.

It should be the chief aim of a university professor to
exhibit humself in hus own true character - that is, as an ignorant
human thinking, actively utilising hus small share of knowledge

6. Possessive and objective cases, "hus" and "hum," fit the
pattern of masculine "his" and "him" rather than feminine
"her" and "her".
-- In "hus" (or "hu's"), -'s is simply a possessive inflection without any
gender bias ("author's, person's, student's, employee's").

As for the objective case, "hum" follows the pattern not
only of "him," but also of the objective pronuns "whom" and "them", which
are gender neutral. "Whom do you prefer?" - "I prefer hum."

I admit, however, that for the objective case the same
form of "hu" can be used as for the nominative. This would
follow the gender neutral "it" where the nominative and
objective cases coincide.

It would clear enough to say:

An introvert can easily become an extravert when it is
advantageous for hu to do so.

Or you can say:

An introvert can easily become an extravert when it is
advantageous for hum to do so.

Thus the four forms of the 3rd person pronouns make up the
table:

----------------------------------------------------------------------------------
nom gen (adj) posses acc refl
--- --- --- --- ----
male he his his him himself
fem she her hers her herself
neut hu hus hus hu huself
(hum)

---------------------------------------------------------------------------------
Excuse me if I have missed any other specific objections.
If you could bring them forth, I would be happy to
consider them.