As a newer chessvariants.com member I'm not sure this is the best place to
post my ideas of what a Next Chess might be like, but here goes. In an old
Chess Federation of Canada blog entry of mine I gave 12 criteria I'd like
in an ideal chess variant or board game of skill. Looking back, many of
these seem too general, but I'll cut & paste them here in case they are of
any interest. Note that at least some criteria are based on subjective
judgement, e.g. I don't think Shogi pieces look as aesthetic as standard
chess pieces, at least in my Western eyes:

1. Computer-resistant (e.g. v. humans not close to best);
2. Any endgame stage not in signif. danger of compromise, e.g. by
tablebases (adjourning poss.);
3. Signif. pop. in N. Amer. & elsewhere (ideally clubs & cash prize
events);
4. Tested (e.g. centuries);
5. Rules not too complex/simple;
6. Has signif. lit. & culture (ideally no licensing/copyright on game);
7. Fixed start position (e.g. aesthetic & not same as chess if variant, or
empty board before 1st player moves);
8. Opening phase not in danger of being played out soon;
9. 2 players usually;
10. Not too many/few pieces or board squares/cells/points & play on 1
board;
11. Pieces look & move nicely & board can be on coffee table (ideally fixed
start position not same as chess if variant);
12. Has mateable kings.

Currently I feel that all 12 of the above criteria cannot ever be even
minimally satisfied by any one chess variant. Sac Chess, a 10x10 variant of
my own creation, was designed to have the potential to one day at least
minimally fulfill all 12 criteria, but the rise of self-teaching computers
(i.e. using neural net techniques) has destroyed my hope that Sac Chess
would be even minimally computer resistant, in terms of a human always
being the best player. My best guess is that aside from a variant that
features changing the rules for how the pieces or game is played every
turn, which would thus be too complex and/or unnatural to play and enjoy
for most people, a very large board size (larger than 19x19) might be the
best hope for a computer resistant variant, but then other of the 12
criteria listed above would not be met.

If we ignore the issue of computer dominance of a game, and the increased
possibilities for cheating that that brings (especially for internet play),
we should assess the other current weak points of chess before deciding if
it needs replacing any time soon, purely in terms of dominance (before we
decide what to replace it with) - chess variants can still co-exist with a
dominant chess for a long time, too.

The other current weak points of chess, besides computer dominance of it,
IMHO are possibly extensive databases, endgame tablebases and the
difficulty top players have in beating each other when using many Black
openings. Are these truly serious weaknesses that threaten the game's
short-term survival, even? I would say no. Yet, like all previous versions
of chess, the immortal game will not live on in its current form forever.
How long does it have left? In a discussion long ago on the Canadian chess
message board chesstalk, I asked International Master Jean Herbert how long
he thought chess might last with its basic rules intact, and after noting
that 'The brand is too strong' he opined it would be at least 500 years
before the death of chess. A game inventor posting on the website, always
on the lookout to push chess variants of all sorts, thought it wouldn't
even last 30 years. I supposed about 100 years would elapse, which is as
good a guess as any IMHO, especially since my bias is less strong either
way.

Even though I don't see the urgency for there to be a Next Chess (barring
future ruinous levels of computer-assisted cheating that may one day wreck
organized board game play in general), I'll now make note of some Game
Courier chess variants (presets) I've thought about playing at some point,
as well as ones that I rejected. It may contribute somehow towards people
deciding someday what to rule in or out as possible Next Chess candidates.
Some of the ones I've thought of playing are more for fun or novelty than
possible replacements for chess, but I generally chose games that were at
least minimally popular, and I much preferred those with rule enforcing
presets (unless I was especially attracted to a variant otherwise).

Games I ruled in for my possible play on Game Courier include chess itself,
Crazyhouse, Glinski's Hexagonal Chess (Symmetric or standard version),
McCooey's Hexagonal Chess (Hexajedrez or standard), Shogi, Chinese Chess,
Alice Chess, Seirawan Chess, Marseillais Chess, Circular Chess, Smess,
Eurasian Chess, Four Kings - Double Mate (Chatarunga) Four Way Chess and Sac Chess.

I found that the above 17 games (many of which quite resemble chess) break
down into 11 types: Drops used, Hexagonal board, Chinese Chess style rules
used, Double move rule used, 10x10 board used, 3D chess, 8x8 & very
chess-like, compound pieces used, Round board used, Smess concept, 2 armies
for each player. Quite a variety of types, which I deliberately steered
for.

Some of the above are not altogether logical games at times, it seems to
me, but their positive sides attract me. Chinese Chess, for example, has a
strange rule concerning the kings not being allowed to see each other on an
open file at long distance, even, but I suppose it's a kludgy rule needed to allow for a decent number of basic checkmates. Eurasian chess is totally
new to me, but I like that only the king can't cross the river, compared to
Chinese Chess, and yet the king still has offensive power due to the same
sort of rule as above.

Now for some notable cases of variants I didn't like so much, in spite of
not rejecting immediately:

Berolina Chess is too close to regular chess, and I didn't like a player
being able to easily open a file with a pawn move;
Korean Chess I found overly complex, and I didn't like a pass rule being
allowed;
Losing Chess doesn't appeal to me in spite of any strategic depth it might
have, since giving away material as a policy seems illogical.
Raumschach is a 3D variant I'd like except it sorely lacks a healthy number
of 'basic mates' - it never occurred to me that there would be such a
fundamental problem with at least some chess variants. Circular chess is
not quite as bad this way, and a round board is a fun concept. With Alice
Chess I had a sound 3D chess variant backup for Raumshach;
Extinction Chess has a similar problem to Losing Chess for me, in that the
concept involved seems unnatural/illogical;
Capablanca Chess didn't appeal to me for 2 reasons: I don't like
rectangular boards, and the Chancellors of both sides could develop
symmetrically and be immediately traded off, though there may be a
strategic reason against this I'm not aware of (Grand Chess has a similar
drawback, and I also mildly don't like the illogic of not being allowed to
promote to a piece type the opponent hasn't captured yet). Seirawan Chess
doesn't have this last drawback, though I would note that, strangely, there
may be less playable openings than in chess (e.g. it may be dangerous to
play an Open Sicilian with White in S-Chess, perhaps);
Honourable mention should go to Rococo, which I barely don't like enough
because so many pieces are queen-like in their range, in spite of differing
capturing methods, and the board's outer ring used only for leaping
captures doesn't seem to make full enough use of a 10x10 board.