The information provided in this answer can lead to insecure programming practices.
The information provided here depends highly on MySQL configuration, including (but not limited to) the program version, the database client and character-encoding used.
MySQL recognizes the following escape sequences.
\0 An ASCII NUL (0x00) character.
\' A single quote (“'”) character.
\" A double quote (“"”) character.
\b A backspace character.
\n A newline (linefeed) character.
\r A carriage return character.
\t A tab character.
\Z ASCII 26 (Control-Z). See note following the table.
\\ A backslash (“\”) character.
\% A “%” character. See note following the table.
\_ A “_” character. See note following the table.
So you need
select * from tablename where fields like "%string \"hi\" %";
Although as Bill Karwin notes below, using double quotes for string delimiters isn't standard SQL, so it's good practice to use single quotes. This simplifies things:
select * from tablename where fields like '%string "hi" %';
You should use single-quotes for string delimiters. The single-quote is the standard SQL string delimiter, and double-quotes are identifier delimiters (so you can use special words or characters in the names of tables or columns).
In MySQL, double-quotes work (nonstandardly) as a string delimiter by default (unless you set ANSI SQL mode). If you ever use another brand of SQL database, you'll benefit from getting into the habit of using quotes standardly.
Another handy benefit of using single-quotes is that the literal double-quote characters within your string don't need to be escaped:
select * from tablename where fields like '%string "hi" %';
For strings like that, for me the most comfortable way to do it is doubling the ' or ", as explained in the MySQL manual:
There are several ways to include quote characters within a string:
A “'” inside a string quoted with “'” may be written as “''”.
A “"” inside a string quoted with “"” may be written as “""”.
Precede the quote character by an escape character (“\”).
A “'” inside a string quoted with “"” needs no special treatment and need not be doubled or escaped. In the same way, “"” inside a
Strings quoted with “'” need no special treatment.