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Batter hits a ball into the left field corner rounds second ...goes quarter way to third sees he can't make it so heads back to third throw almost gets him ...he thinks he is out and walks along the baseline towards first second baseman chases him and he slides into first safely... is there a rule about abandoning a base...

It appears to be a judgement call by the ump as to whether the runner abandoned his efforts because he thought he was out. When they leave the base path to go to the dugout, it is an easy call. Maybe due to their ages, the umps gave him a break. There is no rule against going backwards during a play, but you must touch all the bases in reverse order that they were touched in the first place, or the runner is out.

7.08 Any runner is out when—
(a) (1) He runs more than three feet away from his base path to avoid being tagged
unless his action is to avoid interference with a fielder fielding a batted ball. A
runner’s base path is established when the tag attempt occurs and is a straight
line from the runner to the base he is attempting to reach safely; or
(2) after touching first base, he leaves the base path, obviously abandoning his
effort to touch the next base;
Rule 7.08(a) Comment: Any runner after reaching first base who leaves the base path
heading for his dugout or his position believing that there is no further play, may be declared out if the
umpire judges the act of the runner to be considered abandoning his efforts to run the bases. Even
though an out is called, the ball remains in play in regard to any other runner.
This rule also covers the following and similar plays: Less than two out, score tied last of ninth
inning, runner on first, batter hits a ball out of park for winning run, the runner on first passes second
and thinking the home run automatically wins the game, cuts across diamond toward his bench as
batter-runner circles bases. In this case, the base runner would be called out “for abandoning his effort
to touch the next base” and batter-runner permitted to continue around bases to make his home run
valid. If there are two out, home run would not count (see Rule 7.12). This is not an appeal play.
PLAY. Runner believing he is called out on a tag at first or third base starts for the dugout
and progresses a reasonable distance still indicating by his actions that he is out, shall be declared out
for abandoning the bases.

agree with team mom. judgement call. and sounds more safe than not.

The rule you cite mentions first and third base but not second and the play happened at second....

I believe team mom mentioned it was an umpire judgement call. And have to see it live. it would be more clear if it had been at first or third and he clearly abandon base for dug out. Not being their hard to say, but By your description sounds like safe at first would be fair.

As a former umpire, abandoning a base is seldom (if ever) called. In your description, the runner never left the baseline even though he was running the bases backwards (which isn't illegal). As an umpire I cannot know the runner's intent. Maybe he was trying to decoy the defense? In other words, unless they're almost in the dugout I'm not calling them for abandoning a base.

Per MLB rules, running backwards can be illegal, however, doesnt sound like it would apply in this case:

7.08(i) After he has acquired legal possession of a base, he runs the bases in reverse order for the purpose of confusing the defense or making a travesty of the game. The umpire shall immediately call “Time” and declare the runner out;

Rule 7.08(i) Comment: If a runner touches an unoccupied base and then thinks the ball was caught or is decoyed into returning to the base he last touched, he may be put out running back to that base, but if he reaches the previously occupied base safely he cannot be put out while in contact with that base.